Circular functions. Areas of study Unit 2 Functions and graphs Algebra

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1 Circular functions VCE VCEco coverage reas of stud Unit Functions and graphs lgera In this cha chapter Trigonometric ratio revision B The unit circle C Radians D Smmetr E Identities F Sine and cosine graphs G Tangent graphs H Solving trigonometric equations I pplications

2 0 Mathematical Methods Units and Trigonometric ratio revision Recall that for a right-angled triangle: O = opposite = adjacent H = hpotenuse. These ratios can e memorised using the term SOH CH TO. Eample Find the value of in each of the following triangles. a. c 9 7 H O sin = cos = tan = O --- H --- H O --- Epress lengths correct to decimal places and angles to decimal place. a Write the cosine ratio. a Use cos = --- H Replace with, H with 9 and with. cos = -- 9 Make the suject of the equation. = 9 cos Calculate, correct to decimal places. =.9 O Write the sine ratio. Use sin = --- H Replace O with., H with and with 7. sin 7 = Make the suject of the equation.. = ---- sin 7 Calculate, correct to decimal places. = 9.0 c Write the tangent ratio. O c Use tan = Replace O with 8., with 7. and tan = 7. with. Evaluate the right-hand side of the =. 7 equation, keeping plent of decimal places at this stage. Make the suject of the equation = tan (. 7) using inverse tan notation. Calculate, correct to decimal place, using the inverse tan function of the calculator. = 8.9.

3 Chapter Circular functions 07 ngles of elevation and depression The angle of elevation is an angle aove the horizontal and the angle of depression is an angle elow the horizontal. Eample ngle of depression ngle of elevation Horizontal The shadow of a tree is metres long when the angle of elevation from ground level to the sun is 0. How tall is the tree? Draw a right-angled triangle representing the situation. 7 0 m Let the height of the tree e metres. Let = the height of the tree. Write the tan rule. O Use tan = --- Replace O with, with and with 0. tan 0 = Make the suject of the equation. = tan 0 Calculate, correct to decimal places, using the calculator. = 8. State the solution. The tree is approimatel 8. metres tall. rememer rememer SOH CH TO O O sin = --- cos = --- tan = --- H H H O ngle of depression ngle of elevation Horizontal

4 08 Mathematical Methods Units and Trigonometric ratio revision SkillSHEET. Eample a Find the value of in each of the following, correct to decimal places. a c SkillSHEET d e.9 f g h i... Eample Find the value of in each of the following. a 8 c 7. d e f g h i

5 Chapter Circular functions 09 Eample c Find the value of a in each of the following. a c 8 a. 9 a 0. a..7 d e 8. f. a a.0 a 9 g. h i 0. a 0. a. a multiple choice In the figure elow: a 0 cm.98 cm 0 a c is nearest to: 0 cm B cm C 7. cm D 8 cm E 9 cm angle a is nearest to: B C 7 D E 9 is nearest to: 8 cm B cm C cm D 0 cm E 7 cm multiple choice If sin a = 0.9, then tan a is equal to: B 0.0 C.078 D. E 0.09 multiple choice If 0 < a < 90 and cos a < 0., then which of the following is correct? a < 0 B a > 0 C a < 0 D a < E a > 0 Eample 7 tree which is metres tall casts a shadow so that the angle of elevation from the end of the shadow to the top of the tree is. How long is the shadow? m

6 0 Mathematical Methods Units and 8 How far up a wall will a.-metre ladder reach when it is inclined at an angle of 8 from the wall? 0 m 8 9 o fling a kite on the end of a 0-metre string fies the other end to the ground. How high is the kite when the string makes an angle of 8 with the ground? 0 descending aircraft is on a path which makes an angle of to the ground. What is the inclined distance of the aircraft from the runwa when its height is 00 metres? 00 m Runwa Eample town planner wants to measure the height of a uilding. He lies 0 metres from the ase of the uilding and measures the angle of elevation to the top of the uilding to e 8. How tall is the uilding? 8 0 m mathematicall ale tree removalist measures the angle of elevation of the two points on a diseased section of a large eucalpt at a distance of 0 m from the ase of the tree. If the angles are 0 and respectivel, how tall is the diseased section of the tree? Diseased section 0 0 m

7 Chapter Circular functions 0-metre fling fo cale is set up to cross a river so that it will drop 0 metres verticall. What is the angle of depression of the cale? n oserver on a cliff top which is metres aove the sea spots a whale at an angle of depression of. Find the distance of the whale from the ase of the cliff. soccer plaer has a shot for goal from the position shown in the figure elow. Goal m 7. m m Find the scoring angle,. surveor wishes to determine the height of a radio tower on top of a uilding. t a distance of 80 metres from the ase of the uilding the angles of elevation to the top of the uilding and to the top of the tower are 9 and respectivel. If the radio tower is. m tall, find the value of. 7 Eamine the diagram. a How far is it from to B to C? How far is it from D to E? 0 m C D B E m

8 Mathematical Methods Units and The unit circle Trigonometric ratios can e considered using a unit circle. The position of point P can e descried using the coordinates and and the angle,. The point, P, which is on the circle, is from the positive -ais and can e epressed as P( ). The -coordinate of P is = cosine. The -coordinate of P is = sine. These ma e areviated: = cos = sin The point, P, can also e denoted as P (cos, sin ). The tangent line on a unit circle is the line that is a tangent to the circle at the point (, 0). Tangent (or tan for short) is the height at which a line along an angle hits the tangent line. The diagrams at right show tan for angles in the first and second quadrants. tan O P( ) Note: sin, for all and cos, for all. tan The tangent line sin Note: Tangent (or tan ) ma also e defined as the ratio --. That is, tan =. cos This formula and its derivation will e discussed in more detail later in this chapter. Quadrants The coordinate aes divide the unit circle into four quadrants as shown in the diagram elow (left). The angle measurements, in degrees, etween the quadrants are shown in the diagram elow (right). nd quadrant st quadrant 90 O 80 O 0 or 0 rd quadrant th quadrant 70

9 Chapter Circular functions Therefore: quadrant is where 0 < < 90 quadrant is where 90 < < 80 quadrant is where 80 < < 70 quadrant is where 70 < < 0 Since = sin, sine is positive in quadrants and and sine is negative in quadrants and. Since = cos, cosine is positive in quadrants and and cosine is negative in quadrants and. Since tan = --, tangent is positive in quadrants and and tangent is negative in quadrants and. The diagram summarises which of the trigonometric functions is positive in each quadrant. The word CST will assist in recalling this. You ma also use a short phrase, such as ll Stations To Crodon. Eample If a is an angle, find the quadrant where: a sin a > 0 and tan a < 0 sin a < 0 and cos a > 0. Sine positive Tangent positive ll positive Cosine positive a Sine is positive and tangent is negative in quadrant. Sine is negative and cosine is positive in quadrant. a Quadrant Quadrant Eact values Calculated trigonometric values (sin, cos and tan) of most angles are rational approimations correct to several decimal places. However, for a few particular angles, eact trigonometric values can e determined. These include multiples of 90 and the ratios of isosceles and equilateral triangles. Eact values can e determined for 0 and an multiple of 90 using this diagram and the fact that = sin, = cos and tan = --. The isosceles triangle with equal sides one unit each gives the eact trigonometric values for O 70 The hpotenuse is calculated using Pthagoras theorem: 0 or 0 hpotenuse = + =

10 Mathematical Methods Units and So, using SOH CH TO, sin = cos = tan = -- = = = = The equilateral triangle with each side of length units, shown in the figure at right, has een isected down the middle. 0 Line of isection 0 Using Pthagoras theorem line of isection = = So, using SOH CH TO, sin 0 = -- cos 0 = tan 0 = sin 0 = cos 0 = -- tan 0 = = These eact values are summarised in the tale elow. = ngle ( ) sin cos tan undefined = --

11 Eample Without using a calculator, find: a sin 90 cos 80 c tan 70 d sin 0. Chapter Circular functions a Sine corresponds to the -coordinate on the unit circle. The value of at the point where = 90 is. a sin 90 = Cosine corresponds to the -coordinate on the unit circle. The value of at the point where = 80 is. cos 80 = c d Tangent corresponds to the value of on the unit circle. The value of is and the value of tan 70 = is 0 at the point where = 70. The value of is not defined. tan 70 is undefined 0 Sine corresponds to the -coordinate on d the unit circle. The value of where = 0 is 0. sin 0 = 0 -- c Eample Without using a calculator, state the eact value of: a sin cos 0 c tan 0. a Read the value from the tale, or determine using the special O triangle, sin = ---. H a sin = = = Continued over page

12 Mathematical Methods Units and Read the value from the tale, or determine using the special triangle, cos = ---. H 0 cos 0 = -- 0 c Read the value from the tale, or determine using the special O triangle, tan = c tan 0 = = = 0 rememer rememer On a unit circle:. sin =. cos =. tan = height of point where angle line meets tangent line at RHS of unit circle. tan = --. tan nd quadrant, 80 rd quadrant S T 90 cos C 70 st quadrant P( ) sin 0, 0, th quadrant Special triangles 0 0

13 Chapter Circular functions 7 B The unit circle Eample Without using a calculator, state whether the following values are positive (P) or negative (N). a sin 0 sin c sin 0 d sin 0 e sin 0 f sin 0 g sin h sin i sin j sin ( ) Verif our answers using a calculator. Without using a calculator, state whether the following values are positive (P) or negative (N). a cos 7 cos 8 c cos d cos 00 e cos 0 f cos 9 g cos 0 h cos ( 8 ) i cos ( 0 ) j cos 7 Verif our answers using a calculator. Without using a calculator, state whether the following values are positive (P) or negative (N). a tan tan 7 c tan 8 d tan e tan 99 f tan g tan 08 h tan 00 i tan ( ) j tan ( 7 ) Verif our answers using a calculator. If a is an angle, find the quadrant where: a sin a < 0 and tan a > 0 sin a > 0 and cos a > 0 EXCEL The unit circle Cari The unit circle GC The unit circle Spreadsheet Geometr program multiple choice a If 0 < a < 0, sin a < 0 and cos a < 0, then which one of the following is true? 0 < a < 90 B 90 < a < 80 C 80 < a < 70 D 70 < a < 0 E 0 < a < 80 If 0 < a < 0, sin a > 0 and tan a < 0, then which one of the following is true? 0 < a < 90 B cos a < 0 C 80 < a < 70 D cos a > 0 E 70 < a < 0 c Given that 0 < a < 0, then the equation cos a = has: one solution B no solution C two solutions D three solutions E four solutions d If 0 < a < 0, then the equation sin a = cos a has: no solutions B two solutions C three solutions D one solution E four solutions e If 0 a 80, and sin a = cos a, then a is equal to: 0 B 0 C D 0 E 90 Eample Without using a calculator, find: a cos 80 sin 70 c tan 0 d sin 80 e cos 70 f cos 0 g tan 70 h tan 80 i sin 0 j cos 70 Eample 7 Without using a calculator, state the eact value of: a sin 0 cos c tan 0 d cos 0 e sin 0 f tan g sin h cos 0 i sin 90 j tan 90 k sin 0 l cos 90 WorkSHEET.

14 8 Mathematical Methods Units and Radians You are used to measuring angles in degrees ( ), and will recall that there are 0 in a full circle. n alternative unit for angle measurement is the radian. When drawn at the centre of a circle, a radian is the angle covered one radius length along the circumference of a circle. The shorthand for radian is c. n arc length of one radian, c, is shown on the circle at right. The circumference of the unit circle = r = () = units. So the angle swept in one revolution is equivalent to radians or c. Therefore, c = 0, so: c = 80 c 80 = ---- = -- c 80 These formulas ma e used to convert degrees to radians and vice versa. Eample Convert the following angles into eact radians. a 0 70 radian radius c a Multipl a 0 = 0 80 Simplif dividing through the highest common factor, 0. c Multipl = Simplif dividing through the highest common factor, 0. Eample = = -- c 7 -- c 8 -- c c 80 Convert the following angles to degrees. a --- c 0.8 c a Multipl a 80 = c Cancel out c. = Simplif. = 0 Multipl 0.8 c c = c Cancel out c. = Simplif. = 7

15 Chapter Circular functions 9 When working in degrees, all angles should e epressed to the nearest tenth of a degree unless otherwise stated. Eample Use a calculator to convert: a 7. to radians, correct to decimal places.7 c to the nearest tenth of a degree. a Multipl 7. c --. a 7. = c Evaluate and round off to decimal = c places. 80 Multipl c 80 = Evaluate to decimal place. = 7. 8 Graphics Calculator tip! Degrees and radians Generall, when working with angles on the home screen, Degree MODE is most convenient. When plotting graphs, Radian MODE is sometimes preferale. Be sure to check that our MODE settings are appropriate to the task in hand. Several NGLE functions are found in the nd [NGLE] menu. You ma temporaril override the angle setting using the NGLE menu. For eample, if the MODE setting is Radian, ou ma still work in degrees ensuring the last part of the angle is followed a degrees ( ) sign. Notice that the last eample on the screen at right did not work in degrees, as the final degrees sign was omitted. To convert from radians to degrees:. Set MODE to Degree.. The r sign in the NGLE menu will convert to degrees. (This is not intuitive, ut it works!) To convert from degrees to radians:. Set MODE to Radian.. The sign in the NGLE menu will convert to radians. (gain, this is not intuitive, ut it works!)

16 0 Mathematical Methods Units and Eact values and radians Since c = 80, c c ---- = = = and ---- = 0 So the tale of eact values can e written in terms of radian measurements: c ngle ( c ) sin cos tan c = -- = = undefined c c 0 c 0 0 c c 0 c c c 0 c 7 c 0 c c 0 c c Other important angles are shown on the circle on the right. 00 c 0 c 7 c ( sectors shaded) Our special triangles can e updated to include radians as shown. 0 0

17 Chapter Circular functions Eample Find the eact value of: a sin -- tan a Read the eact value from the tale or use a special triangle, O sin = ---. H 0 a sin -- = 0 Read the eact value from the tale or use a special triangle, O tan = tan -- = or 0 rememer rememer. c = 80. c 80 = = -- c 80 Special triangles can e used to otain eact values for sin, cos and tan. 0 0

18 Mathematical Methods Units and C Radians GC program Degrees and radians Eample Convert the following angles into eact radians. a 0 c 0 d 0 e 0 f 90 g 70 h 0 i 0 j k 0 l 00 m 8 n 7 Mathcad Degrees and radians Eample 7 Convert the following angles to degrees. a c d -- 9 e. f -- g -- h 0.7 multiple choice The smallest angle measurement listed elow is: 8 B C.9 c D E multiple choice The largest angle measurement listed elow is: B C 0 D E.8 c multiple choice If sin a = cos a, then a could e equal to: -- B -- C 0 D -- E Eample 8a Eample 8 Use a calculator to convert the following angles to radians, correct to decimal places. a c d 9. e. f 7. g 70. h.8 7 Use a calculator to convert the following radian measurements to degrees, to the nearest tenth of a degree. a 0..7 c. d 0.98 e. f.7 g.0 h 8 Eample 8 Without using a calculator, find the eact value of each of the following. 9 a sin -- cos -- c tan -- d sin e tan -- f cos -- g sin -- h cos -- --

19 Chapter Circular functions Smmetr The unit circle can e divided into smmetrical sections, as shown in the P( ) diagram on the right. Relationships etween the circular c functions sine, cosine and tangent can e estalished, ased on these smmetrical properties. P( + ) For simplicit, assume is an acute angle, although the following properties hold for an. Quadrant s alread seen: Quadrant B smmetr: Quadrant B smmetr: Quadrant B smmetr: sin = cos = tan = -- sin ( ) = = sin cos ( ) = = cos tan ( ) = = tan sin ( + ) = = sin cos ( + ) = = cos tan ( + ) = = tan sin ( ) = = sin cos ( ) = = cos tan ( ) = = tan Notes. These relationships also appl if degrees are used in place of radians that is, if is replaced 80 or is replaced 0.. n angle measurement is assumed to e in radians unless the degree smol is given. c P( ) = (cos, sin ) = (, ) c S 0 or c P( ) The unit circle EXCEL The unit circle Cari The unit circle Unit circle smmetr Mathcad Spreadsheet Geometr Mathcad Provided an angle is epressed as ± or ±, the trig function (sin or cos) remains the same, onl the sign (+ or ) ma change. Use this diagram to determine the sign. T C

20 Mathematical Methods Units and Eample 0 a If sin = 0.9, find sin (80 + ). If cos = 0., find cos (0 ). c If tan =.7, find tan ( ). d If cos = 0.8, find cos ( + ). a Sketch the angle on a unit circle and relate it to the first quadrant. a 80 sin (80 + ) (80 + ) S T C sin B smmetr sin (80 + ) = sin. sin (80 + ) = sin Replace sin with 0.9. = 0.9 Sketch the angle on a unit circle and relate it to the first quadrant. S T C cos cos (0 ) (0 ) c B smmetr cos (0 ) = cos. cos (0 ) = cos Replace cos with 0.. = 0. c Sketch the angle on a unit circle and relate it to the first quadrant. S T C tan tan ( ) d B smmetr tan ( ) = tan. tan ( ) = tan Replace tan with.7. =.7 Sketch the angle on a unit circle and relate it to the first quadrant. d cos ( + ) S + T cos C B smmetr cos ( + ) = cos. cos ( + ) = cos Replace cos with 0.8. cos ( + ) = 0.8

21 Eample Chapter Circular functions Find, without using a calculator, the eact value of each of the following. 7 a tan 0 sin 0 c cos d tan a Epress tan 0 as tan (80 0). a tan 0 = tan (80 0) Sketch the angle on a unit circle and relate it to the first quadrant. (80 0) S T C 0 tan 0 = tan (80 0) B smmetr tan (80 0) = tan 0. tan 0 = tan 0 Replace tan 0 with its eact value, or. = or Epress sin 0 as sin (0 0). sin 0 = sin (0 0) Sketch the angle on a unit circle and relate it to the first quadrant. S T 0 C sin 0 sin (0 0) (0 0) B smmetr sin (0 0) = sin 0. sin 0 = sin 0 Replace sin 0 with its eact value, --. = -- c Epress cos as cos. c cos = cos Sketch the angle on a unit circle and relate it to the first quadrant. cos ( ) S cos T C Continued over page

22 Mathematical Methods Units and B smmetr cos = cos. -- cos = cos Replace cos -- with its eact value of or = or. d Epress tan 7 as tan. d tan = tan Sketch the angle on a unit circle and relate it to the first quadrant. + S T C tan = tan ( + ) B smmetr tan + = tan. -- tan = tan Replace tan -- with its eact value or. = or Eample If sin = 0.9, evaluate each of the following (without using a calculator). 8 a sin sin a Epress sin as sin = sin. a sin = sin Using smmetr epress sin as sin. = sin Replace sin with 0.9. = Epress sin -- as sin = sin. sin = sin B smmetr epress sin as sin. = sin Replace sin with 0.9. = 0.9 8

23 Chapter Circular functions 7 rememer rememer. unit circle sketch of the given angle related ack to the first quadrant is often helpful.. Provided an angle is epressed as ± or ±, the trig function (sin or cos) remains the same, onl the sign (+ or ) ma change.. Use this diagram to determine the sign.. ma e written as , -----, -----, etc. 8 S T C D Smmetr Eample 0 If sin = 0., find: a sin (80 ) sin (80 + ) c sin ( ) d sin (0 ). If cos = 0., find: a cos ( ) cos ( + ) c cos ( ) d cos ( ). EXCEL The unit circle Spreadsheet Eample c, d Eample If tan =., find: a tan ( ) tan ( ) c tan ( + ) d tan ( ). Given that sin a = 0., cos = 0.7 and tan c = 0.9, write down the value of each of the following. a sin (80 + a) cos (80 ) c tan (0 c) d sin ( a) e sin (80 a) f cos ( ) g cos (0 ) h tan (80 c) i tan (80 + c) Find, without using a calculator, the eact value of each of the following. a sin 0 cos c tan 0 d tan 0 e sin 0 f cos 0 g tan 0 h sin 00 i cos ( 0) j sin ( ) k tan 80 l sin 70 Find, without using a calculator, the eact value of each of the following. 7 a cos sin c tan d sin e cos f tan g sin h cos i tan -- j sin k cos l sin -- 7 If sin -- = 0.8, cos -- = 0.9 and tan -- = 0., evaluate each of the following (without using a calculator) a sin cos c tan d cos -- 7 e sin f tan Cari The unit circle Unit circle smmetr The unit circle Geometr Mathcad Mathcad

24 8 Mathematical Methods Units and 8 Given that sin 7 = 0.9, cos 7 = 0.9 and tan 7 =.7, find the value of each of the following (without using a calculator). a sin 0 cos c tan 8 d sin e cos f tan ( 7) 9 If sin 0.7 = 0., cos 0.7 = 0.7 and tan 0.7 = 0.8, find the value of each of the following, without using a calculator. (Hint: =., approimatel.) a sin. cos.8 c tan.8 d sin ( 0.7) Career profile BRONWYN LYCOCK Scientist Contact lenses are used to alter the path of light through the pupil so that it focuses clearl on the retina. The ending of light is called refraction. For a ra of light travelling from air to another material, the refractive inde of the material can Qualifications: BSc, PhD (Chemistr), Grad Dip (Soil science) sin e calculated using the formula: n = - i, sin r where i is the angle of incidence, r is the angle of refraction and n is the refractive inde. Emploer: CSIRO Compan wesite: i r ir Surface of material The project I have een working on, as part of a team, involves producing a contact lens that can e left in the ee for 90 das. To sta in place for such a length period, a lens has to e soft, comfortale, allow ogen to pass through to the ee surface and not cause an ill effects. M contriution to the project is to design and produce new polmers from which the lenses can e manufactured, and which, hopefull, will have the required properties. M da is alwas varied. It usuall involves laorator work snthesising new materials; adding other components to these materials to give them the desired properties; moulding the materials into lenses; testing the lenses for water content and impurities; and the inevitale cleaning up. I keep up with developments around the world reading journals in m area of epertise. Material If an two of the quantities are known, the third can e calculated using this formula (known as Snell s Law). Questions. Calculate the refractive inde of a particular plastic if a light ra enters the plastic with an angle of incidence of and has an angle of refraction of.. ra of light strikes a glass lock of refractive inde.7 at an angle of incidence of 0. What is the angle of refraction?. Find out wh light ends as it travels from one material (or medium) to another.

25 Chapter Circular functions 9 Identities n identit is a relationship that holds true for all legitimate values of a pronumeral or pronumerals. For eample, a simple identit is + =. The identities descried in this section are far more interesting and useful than this, as ou will see. The Pthagorean identit Consider the right-angled triangle in the unit circle shown. ppling Pthagoras theorem to this triangle gives the identit: sin + cos = O cos sin D P( ) The tangent Consider the unit circle on the right. tangent is drawn at and etended to the point C, so that OC is an etension of OP. This tangent is called tangent, which is areviated to tan. Triangles ODP and OC are similar, ecause the have their three corresponding angles equal. O cos P( ) sin D B C tan or tan sin It follows that: = (corresponding sides) cos tan = sin cos (as mentioned in an earlier section). nother relationship etween sine and cosine complementar functions Consider the unit circle shown on the right: The triangles OB and ODC are congruent ecause the have all corresponding angles equal and the hpotenuse equal (radius = ). Therefore all corresponding sides are equal and it follows that: sin (90 ) = cos = and cos (90 ) = sin = OR sin (-- ) = cos and cos (-- ) = sin D C(90 ) O B( ) We sa that sine and cosine are complementar functions. Though not required for this course, ou ma like to tr to find the complementar function for tangent, that is, tan (90 ) =?

26 0 Mathematical Methods Units and Eample If sin = 0. and 0 < < 90, find, correct to decimal places: a cos tan. a Use the identit sin + cos =. a sin + cos = Sustitute 0. for sin. (0.) + cos = Solve the equation for cos correct to decimal places. cos = 0. = 0.8 cos = ± 0.8 = 0.97 or 0.97 Retain the positive answer onl as cosine is positive in the first quadrant. For 0 < < 90, cos is positive so cos = 0.97 sin sin Use the identit tan =. tan = cos cos Sustitute 0. for sin and 0.97 for cos. Calculate the solution correct to decimal places. Eample = = 0. Find all possile values of sin if cos = 0.7. Use the identit sin + cos =. sin + cos = Sustitute 0.7 for cos. sin + (0.7) = Solve the equation for sin correct to decimal places. sin = 0. = 0.7 sin = ± 0.7 Retain oth the positive and negative solutions, since the angle could e in either the first or fourth quadrants. = 0. or 0. Eample Find a if 0 < a < 90 and a sin a = cos cos a = sin 7. a Write the equation. a sin a = cos Replace cos with sin (90 ) (complementar functions). sin a = sin (90 ) sin a = sin 8 a = 8 Write the equation. cos a = sin 7 Replace sin 7 with cos (90 7). cos a = cos (90 7) cos a = cos 7 a = 7

27 Eample If 0 < a < 90 and cos a = --, find the eact values of: a sin a tan a c cos (90 a) d sin (80 + a). Chapter Circular functions Draw a right-angled triangle. Mark in angle a, its adjacent side () and the hpotenuse (H). H = a = Use Pthagoras to calculate the opposite side (O) to a. O = O = = O = O a Use the right-angled triangle to find ---. a sin a = H Sustitute O = and H =. = O Use the right-angled triangle to find ---. tan a = Sustitute O = and =. = c Use the identit cos (90 a) = sin a. c cos (90 a) = sin a d Sustitute sin a =. cos (90 a) = Use the smmetr propert sin (80 + a) = sin a. d sin (80 + a) = sin a Sustitute sin a =. sin (80 + a) = (Note: The aove results could have een otained using the identities directl.) O --- H O --- rememer rememer. sin + cos =. sin (90 ) = cos. sin -- = cos sin. tan =. cos (90 ) = sin. sin -- = cos cos

28 Mathematical Methods Units and E Identities Cop and complete the tale elow, correct to decimal places: sin cos sin + cos Eample a Eample Eample If sin = 0.8 and 0 < < 90, find, correct to decimal places: a cos tan. If cos = 0. and 0 < < 90, find, correct to decimal places: a sin tan. Find all possile values of the following. a cos if sin = 0. cos if sin = 0.7 c sin if cos = 0. d sin if cos = 0.9 Use the diagram at left to find the eact values of: a c sin c cos. c Use the diagram at right to find the eact values of: a cos c tan Find the eact values of: a cos if sin = and 90 < < 80 sin if cos = -- and is in the third quadrant 7 c cos if sin = and is in the fourth quadrant d sin if cos = and < < 8 multiple choice a Eamine the diagram at right and answer the following questions. a sin is equal to: cos B cos C tan D sin E tan cos is equal to: tan B cos C tan D sin E sin c

29 Chapter Circular functions c d tan is equal to: cos sin B sin cos C D sin cos E sin + cos tan is equal to: cos sin B sin cos C D sin cos E sin cos sin cos cos sin Eample 9 Find a if 0 a 90 and: a sin a = cos 0 sin a = cos 8 c cos a = sin 9 d cos a = sin 8 e sin 8 = cos a f cos = sin a g sin 89 = cos a h cos 7 = sin a. 0 Cop and complete the following tale. sin cos tan Eample If 0 < a,, c < 90 and sin a =, cos = --, tan c = ---, find: a sin tan c cos a d tan a e sin c f cos c g sin (90 a) h cos (90 ) i sin (90 c) j sin (80 a) k cos (80 + ) l tan (80 + c). Further trigonometric identities sin The equations tan = and sin + cos = are not the onl non-trivial cos trigonometric identities. Prove (or at least verif) that the equations elow are also identities using one of the following methods: (i) Use the identities aove, and algeraic manipulation. (ii) Complete a tale of values for several values of and show that the left side of the equation equals the right side. (iii) Plot the left-hand side as Y and the right hand side as Y using a graphics calculator (or using graphing software) to show oth sides graphs are identical. sin = sin cos sin + sin = sin cos + tan = -- cos sin ( + ) = sin cos + cos sin

30 Mathematical Methods Units and Sine and cosine graphs The graph of = sin To get an idea of what the graph of = sin looks like, we ma first construct a tale of values = sin Net, these values are plotted on a set of coordinate aes and a smooth curve is drawn to join the points. (, 0) (0, 0) 0 (, 0.7) (, ) (, 0.7) (, ) (, 0.7) (, 0.7) (, 0) (, ) = sin (, 0) 7 (, 0.7) Verif the shape of this curve using a graphics calculator. It can e oserved that the curve repeats itself in ccles after an interval of units. Due to this repetition it is called a periodic function and the period is the interval etween repetitions. The period of = sin is radians (or 0 ). When dealing with graphs of circular functions such as sin, unless otherwise stated, we assume that the units for are radians. The mean position of the curve is = 0 and the maimum and minimum values are and respectivel. The distance from the mean position to the maimum (or minimum) position is called the amplitude of the periodic function. The amplitude of = sin is unit. Period mplitude Mean position Period The graph of = cos s for the graph of = sin, the graph of = cos can e estalished first comleting a tale of values = cos

31 Chapter Circular functions When these points are plotted on a set of aes and joined with a smooth curve, the graph looks like this: Verif the shape of this curve using a graphics calculator. It can e seen that the asic shape of the curve = cos is the same as that of = sin. It is also a periodic function with: period = amplitude =. Note: The graph of = cos is eactl the same as that of = sin translated (radian) or 90 to the left. (, 0.7) (, 0) 0 (, 0.7) (, ) (, 0.7) Sine and cosine graphs Use a graphing package (such as Mathcad, Graphmatica, GrafEq, MathView or DERIVE ) or a graphics calculator to investigate the following graphs. In each case, sketch the graph from the screen, and state the amplitude and period. Repeat question, replacing sin with cos. If the general forms of the aove graphs are = a sin + c and = a cos + c, descrie the effect of a, and c on the graph. (0, ) (, 0.7) (, 0) 7 (, 0.7) (, 0) a = sin = sin c = sin (, 0.7) (, ) d = sin e = sin f = sin g = sin h = sin -- i = sin j = sin + k = sin l = sin m = sin + = cos (, ) -- units The following generalisations can e made for the graphs of sine and cosine functions. Circular function Period mplitude = a sin = a cos (or ---- ) a 0 (or ---- ) a where a, R and a means the size or magnitude of a, epressed as a positive numer, as the amplitude must alwas e positive. If a < 0 then the resulting graph is a reflection in the -ais of the graph for which a > 0. For eample, the graph of = sin is the reflection of = sin in the -ais (think of the -ais as a plane or flat mirror).

32 Mathematical Methods Units and Eample 7 State i the period and ii the amplitude of each of the following functions. a =. sin c = cos -- 0 a From the graph the ccle repeats after units. a ii Period = From the graph, the distance from the mean position to the maimum position is units. ii mplitude = Write the formula for the period of = a sin. ii Period = where = Sustitute =. ii Period = Simplif. ii Period = -- B rule, the amplitude is a or.. ii mplitude =. ii mplitude =. c Write the formula for the period. c ii Period = where = -- Sustitute = --. ii Period = Simplif. ii Period = B rule, the amplitude is. ii mplitude = -- Eample Sketch the graphs of the following functions and state i the period and ii the amplitude of each. a = cos -- [0, ] = -- sin [0, ] a Write the formula for the period. aiperiod = Sustitute =. -- = Simplif the value of the period =

33 Chapter Circular functions 7 The amplitude is the value in front of cos, written as a positive value. Draw a set of aes. Since the amplitude is, mark or imagine horizontal guidelines at = and. Sketch one ccle of the graph ever period (ever ) along the -ais, for [0, ], showing ke -values. 7 ii mplitude = Write the formula for the period. ii Period = Sustitute =. ii Period = Simplif the value of the period. ii Period = -- The amplitude is the value in front of sin, ii mplitude = written as a positive value. Draw a set of aes. Since the amplitude is --, mark or imagine horizontal guidelines at = -- and --. Sketch one ccle of the graph ever period 7 (ever -- ) along the -ais, for [0, ]. 0 8 Determine the -intercepts and mark these on the graph. -- (= ) 7 Eample 9 Sketch the graph of the following function. f: [, ] R, f () = cos -- The period of the function is -----, Period = where = Simplif the value of the period. = Continued over page

34 8 Mathematical Methods Units and The amplitude is. mplitude = Draw a set of aes. Using an interval of (= period), mark the -ais from to (the specified domain). Show and on the -ais, since the amplitude is. = cos Visualise, or check, the general shape of the graph of = a cos. Starting from the point (0, ) complete one ccle of the cosine function forward to (, ). Complete half of a ccle ack from (0, ) to (, ). The -intercepts are halfwa etween the maimum and minimum points. Verif that this graph is correct using a graphics calculator. rememer rememer. Basic graph tpes (a) = a sin = a cos () Period = -----, amplitude = a = a sin a a = a cos a 0 = period a 0. Sketching (a) Recall asic graph tpe. () Find period and amplitude. (c) Sketch in sections of one period. (d) Find -intercepts etween other known intercepts or minimum and maimum points.

35 Chapter Circular functions 9 F Sine and cosine graphs Eample 7a State i the period and ii the amplitude for each of the following functions: a c. Sine graphs Mathcad 0 d e f 0 0. EXCEL Sine graphs Spreadsheet Eample 7, c g h State i the period and ii the amplitude of each of the following functions. a = sin = sin c = sin d = sin e = -- sin -- f = cos g = 0. cos -- h = cos i =. cos j = sin k = -- sin cos l = Cosine graphs EXCEL Cosine graphs GC Trigonometric graphs Cari Sine and cosine graphs Mathcad Spreadsheet program Geometr Eample 8 Sketch the graph of the following functions for [0, ] and state i the period and ii the amplitude of each. a = sin = cos c = sin d = cos e = -- sin f = -- cos g = sin -- h = cos -- i = cos j = sin Check our answers using a graphics calculator

36 0 Mathematical Methods Units and multiple choice Parts a to c refer to the graph. 0 a c The amplitude of the function is: -- B C D E The period of the function is: B -- C D E The equation of the function could e: = sin B = sin C = cos D = sin E = sin -- multiple choice 0 The equation of this curve could e: = cos B = sin -- C = cos D = cos -- E = cos -- State the equation of each of the functions graphed elow. a. 0. 0

37 Chapter Circular functions c d 0 0 e f Sketch the graph of each of the following functions. a f() = cos for [, ] f() = cos -- for [, ] c f() = sin for [0, ] d f() = cos for [0, ] e f() =. sin -- for [, ] f f() = sin for [0, ] Check our graph using a graphics calculator. Eample 9 8 Sketch the graphs of each of the following functions. Check our graphs using a graphics calculator. a f: [, ] R, f() = -- sin f: [, ] R, f() =.8 cos c f: [0, ] R, f() =. sin d f: [0, 8] R, f() = cos For each of the functions graphed elow, state the rule using full function notation. a c 0 8 f() f() 0 f() 0 d e f.8 f() f().. 0 f() WorkSHEET.

38 Mathematical Methods Units and Tangent graphs The graph of = tan To manuall plot the graph of = tan, the following tale of values ma e used = tan 0 undefined 0 undefined 0 undefined 0 (Note: Multiples of -- could e used to give more points to plot and a clearer indication 8 of the shape of the graph.) sin Note the presence of some undefined -values. This is ecause = and cos cos = 0 at these values. These undefined values are shown as vertical asmptotes through the given value of for which the occur. (n asmptote is a line that a graph approaches, ut never quite reaches. In the case of = tan, approaches (ut never actuall reaches) and + for particular -values.) The graph of = tan is shown in the following figure. Verif this graph using a graphics calculator. The features of the graph of = tan are:. It has vertical asmptotes though =... --, --, -----,.... It has no amplitude.. It has a period of.. It has a range of R (the set of all Real numers). (, ) (, 0) Tangent graphs Use a graphing package such as Mathcad, Graphmatica, GrafEq, MathView or DERIVE or a graphics calculator to investigate the following graphs. In each case, sketch the graph from the screen, and state the period. Show all graphs etween 0 and. If the general form of the aove graphs is = a tan, descrie the effect of a and on the graph. Vertical asmptotes ( (, ), ) (0, 0) (, 0) (, 0) 0 7 (, ) (, ) a = tan = tan c = tan d = tan e = tan f = tan g = tan h = tan -- i = tan = tan

39 Chapter Circular functions In general, the graph of = a tan has the following properties:. No amplitude. Period = --. -intercepts at = ± -- and ever period to the left and right of these. smptotes at = and = and ever period to the left and right of these. The following formula for asmptotes applies: ( n + ) asmptote = ± -----, where n = 0,,,... 0 Eample For each function elow, state i the period and ii the equation of the two asmptotes closest to the -ais. a = -- tan = tan -- a i Use the formula for the period. a i Period = -- Sustitute =. = -- ii Use the formula to find the two ii smptotes: = ± closest asmptotes to the -ais. Sustitute =. = ± Simplif. = ± -- 8 i Use the formula for the period. i Period = -- Sustitute = --. = -- ii 0 Simplif. = Use the formula to find the two ii smptotes: = ± closest asmptotes to the -ais. Sustitute = --. = ± Simplif. = ± --

40 Mathematical Methods Units and Eample Sketch the graph of the function = tan over [, ] Find the period using = --. Period = = Find the two asmptotes closest to the smptotes: = ± -ais sustituting = -- into -- = ± = ± State the other asmptotes adding/ Other asmptotes are: =, = +, sutracting the period, units, to/from = + + etc. the first asmptotes, = ±. State all asmptotes in the domain ll asmptotes in the domain [, ] are: [, ]. =, =, =. Evaluate when = -- and = -- to When = --, = tan -- estalish two definite points on the = graph. = When = --, = tan -- Draw a set of aes using [, ]. Mark in the vertical asmptotes at =, =, =. Using the asmptotes as a guide, sketch the standard tan curve. Verif that this graph is correct using a graphics calculator. 0 = = = tan rememer rememer For = a tan :. Period = No amplitude. smptotes at = ± -----, and ever period to the left and right of these.

41 Chapter Circular functions G Tangent graphs Eample 0 State i the period and ii the equation of the two asmptotes closest to the -ais for each of the following. a = tan = tan c = tan -- d = tan -- e = tan f = tan -- g = tan h = tan -- Sketch the graph of each function in question, showing the first two ccles. multiple choice The function = tan has a period equal to: B -- C D -- E multiple choice Use the graph to answer questions a and. Tangent graphs EXCEL Tangent graphs GC Trigonometric graphs Mathcad Spreadsheet program. 0 a The period of the function is equal to: -- B C -- D E The equation of the function is: =. tan B =. tan -- C =. tan D =. tan E = tan Eample Sketch the graphs of each of the following functions over the given domain. a = tan --, [0, ] = tan -----, [0, ] c = tan --, [0, ] d =.8 tan --, [0, ] e = tan, [, ] There is etension material availale on the Maths Quest CD-ROM for further work on sine and cosine graphs. etension Further trigonometric graphs

42 Mathematical Methods Units and Solving trigonometric equations To find the solution to the equation sin = where [0, ] we can consider the graph of = sin. B drawing a horizontal line through =, it can e seen that there are four solutions in the domain [0, ]. The solution for 0 < < --, that is, in the first quadrant, is -- (from our knowledge of eact values). Note: For ineact solutions in the first quadrant use a calculator. 0 7 = sin sin ( ) S sin T C The sine function is also positive in the second quadrant. Using sine smmetr, the net solution is -- = (different smmetr properties are used for cosine and tangent). Since the graph is periodic, an further solutions are found adding (or sutracting) the period () to (or from) each of the first two solutions. For eample, two further solutions are: -- + and = and -- 9 Therefore, four solutions in the specified domain are --, -----, and --. When solving trigonometric equations, the following need to e determined.. The first quadrant angle, irrespective of the sign.. The two quadrants in which the given function is positive or negative.. Two solutions etween = 0 and = (use the appropriate sine, cosine or tangent smmetr propert). If more solutions are required:. Repeatedl add (or sutract) the period to the two solutions as man times as required, noting solutions after each addition or sutraction.. Stop when all solutions within the specified domain are found.

43 Eample Chapter Circular functions 7 Find to the nearest tenth of a degree if cos = 0.8 and [0, 0 ]. Write the equation. cos = 0.8 Find the first quadrant angle solving cos = 0.8 to the nearest tenth of a degree. First quadrant angle = cos 0.8 =. Identif where the cosine function is Cosine is negative in quadrants and. negative. Use cosine smmetr to find the solutions. (80.) 80 (0.8) (80 +.) = (80.) or (80 +.) Simplif the solutions. =. or. Since the period is 0, no further solutions are required for [0, 0 ]. Eample If [0, ], find: a solutions for correct to decimal places if sin = 0. eact solutions to cos =. a Write the equation. a sin = 0. [0, ] Use a calculator to find the first First quadrant angle = sin 0. quadrant angle. First quadrant angle = 0.0 For eample, on a TI8 graphics calculator ensure MODE is set to Radians, and press nd [sin - ] 0.. Identif where sine is positive. Sine is positive in quadrants and. Continued over page

44 8 Mathematical Methods Units and Use sine smmetr to find the two solutions for [0, ]..87 c.0 c For [0, ] = 0.0 or ( 0.0) = 0.0 or.87 Convert the specified domain to a decimal. [0, ] = [0,.] dd (=.8) to each of the solutions aove. dding (=.8) to the last two solutions would give solutions eond the specified domain, so stop here. For [0, ] = 0.0,.87, ( ), = ( ) = 0.0,.87,.88, 9.0 Write the question. cos = [0, ] Recall a special triangle to find the first First quadratic angle = cos - quadrant angle. First quadratic angle = -- (Note: This is not a solution in this case.) Identif where cosine is negative. Cosine is negative in quadrants and. Use cosine smmetr to find the two solutions for [0, ]. dd to each of the solutions aove for [0. ]. dding (= ) to the last two solutions would give solutions eond the specified domain ( or -- ), so stop here. For [0, ] = -- or + -- = or For [0, ] = -----, -----, , = -----, -----, , = -----, -----, -----, --

45 Chapter Circular functions 9 Eample Find if sin = 0.98 and [0, ]. Write the equation. sin = 0.98 [0, ] Divide oth sides to get sin itself. sin = 0.9 Determine the first quadrant angle First quadrant angle = sin 0.9 using a calculator in RD mode, = 0. correct to decimal places. Identif where the sine is positive. Sine is positive in quadrants and. Use sine smmetr to find the solutions. ( 0.) c c 7 = 0. or ( 0.) Simplif the solutions. = 0. or.8 Since the period is, no further solutions are required over the domain [0, ]. Note: n equation not in the form sin = B (or cos or tan) should e transposed efore the solutions are found. rememer rememer. When solving trigonometric equations, the following need to e determined. (a) The first quadrant angle, irrespective of the sign. S () The two quadrants in which the given function is positive or negative. (c) Two solutions etween = 0 and = (use the appropriate sine, cosine or tangent smmetr propert).. If more solutions are required: T C (a) Repeatedl add (or sutract) the period to the two solutions as man times as required, noting solutions after each addition or sutraction. () Stop when all solutions within the specified domain are found.

46 0 Mathematical Methods Units and H Solving trigonometric equations EXCEL Spreadsheet Mathcad Mathcad Mathcad Trigonometric equations Solving sine equations Solving cosine equations Solving tangent equations Eample Eample Eample Find the eact value of in terms of in each of the following equations, given that is in the first quadrant. a cos = sin = -- c tan = Find the value of to the nearest tenth of a degree in each of the following equations, given that [0, 0 ]. a sin = 0. cos = 0. c tan =.7 d sin = 0.8 e cos = 0.9 f tan = 0.87 g sin = 0. h cos = 0.77 Find the value of in each of the following equations if [0, ]. Give answers correct to decimal places, unless eact answers ma e found. a sin = 0.8 cos = 0. c tan =. d tan = e sin = 0.9 f cos = Find eact solutions to each of the following equations over the domain [0, ]. a sin = cos = c tan = d sin + = 0 e cos + = 0 f tan + = 0 d cos = e tan = f sin = multiple choice a If sin = cos = and 0 < < 0, then is equal to: 0 or 0 B or C D or E 0 Given that tan = and 0 < < --, then is equal to: -- B -- C -- D -- E multiple choice The solution to the equation: cos + = 0, over the domain [0, ] is: 7 --, B -----, C --, D -----, E -----, 7 Solve each of the following, to the nearest tenth of a degree, over the domain [0, 0 ]. a sin = cos = c tan 7 = 0 d + sin = e + cos = f tan + 9 = 0 8 Challenge: Solve the following equations over [0, 0 ]. a sin ( + 0 ) = cos ( 0 ) = 0 c tan ( + ) = d sin ( 0 ) = e cos ( + 90 ) + = 0 f sin = 0 g cos -- = h sin -- = i tan = j cos = k tan ( 0 ) = l sin -- ( 0 ) =

47 Chapter Circular functions pplications Man situations arise in science and nature where relationships etween two variales ehiit periodic ehaviour. Tide heights, sound waves, io-rhthms and ovulation ccles are eamples. In these situations trigonometric functions can e used to model the ehaviour of the variales. The independent variale () is often a measurement such as time. When modelling with trigonometric functions ou should work in radians unless otherwise instructed. Eample E. coli is a tpe of acterium. Its concentration, P parts per million (ppm), at a particular each over a -hour period t hours t after am, is descried the function: P = 0.0 sin a Find the i maimum and ii minimum E. coli levels at this each. What is the level at pm? c How long is the level aove 0. parts per million during the first hours after pm? t a Write the function. a P = 0.0 sin t ii The maimum value of the sine ii The maimum P occurs when sin =. function is. ii t Sustitute sin = into the equation for P and evaluate. State the solution. The minimum value of the sine function is. t Sustitute sin = into the equation for P and evaluate. State the solution. Ma. P = 0.0() + 0. = 0. The maimum E. coli level is 0. ppm. ii The minimum P occurs when t sin =. Min. P = 0.0( ) + 0. Min. P = 0.0 The minimum E. coli level is 0.0 ppm. t pm it is 9 hours since am. t pm, t = 9. 9 Sustitute t = 9 into the equation for P, When t = 9, P = 0.0 sin and evaluate. = 0.0 sin = = = 0. State the solution. The E. coli level at pm is approimatel 0. ppm. Continued over page

48 Mathematical Methods Units and c sketch graph will give a etter understanding of this question. c State the amplitude. mplitude = 0.0 Calculate the period. Period = = Identif the asic graph. t The asic graph is P = 0.0 sin State the translations needed. No horizontal translation. Vertical translation of 0. unit up. Sketch the graph of P. P(parts per million) 7 Draw a horizontal line through P = t(hours) Identif where P > 0. from the graph. Solve the equation P = 0. to find the first two values of t. Find the difference etween the solutions t = and t = 0. State the solution. The graph shows that P > 0. etween the first two points where P = 0.. When P = 0., t 0.0 sin = 0. t 0.0 sin = 0.0 t sin = 0. t = -- or -- = -- or t = -- or -- t = or 0 P > 0., for 0 = 8 hours The E. coli level is aove 0. parts per million for 8 hours.

49 rememer rememer Chapter Circular functions. General equations: = a sin + c, = a cos + c,. Period = -----, amplitude = a.. To find maimum value of a function, replace sin or cos with +.. To find minimum value of a function, replace sin or cos with.. Initial values occur at t = 0.. Sketch the graph for greater understanding. I pplications The weight of a rait over a period of time is modelled the graph. a State i the amplitude and ii the period. Epress W as a function of t. W (kg) 0 t (das) EXCEL Trigonometric equations Spreadsheet Mathcad The diagram shows the heart rate of an athlete during a particular hour of a workout. a Find the initial heart rate. State i the amplitude and ii the period. c Epress H as a function of t. H (eats/min) t (min) Solving sine equations Solving cosine equations Mathcad Eample The height aove the ground, h metres, of a child on a swing at an time t seconds after eing released is: t h = + 0. cos Find: a the maimum height of the swing the height after i seconds and ii -- seconds c the length of time that the swing is elow. metres, travelling from one side to the other. The temperature, T ( C), inside a uilding on a given da is given the function: t T = 8 sin where t is the numer of hours after 8 am. a What is the maimum temperature in the uilding and the time at which it first occurs? Find the temperature at i 8 pm, ii pm, iii am (midnight).

50 Mathematical Methods Units and The displacement, (in mm), of a harp string t seconds after it is initiall plucked is modelled the function: (t) = sin 0t. a What is the i amplitude and ii period of this function? How man virations (that is, ccles) will it complete in one second? c Find the displacement after 0.08 seconds. d t what time will its displacement first e mm? The height of a ungee jumper, h metres, aove a pool of water at an time t seconds after jumping is descried the function: h(t) = 0 cos 0.8t + 0 a What is the initial height of the ungee jumper? c When, if at all, does the ungee jumper first touch the water? ssuming the cord is perfectl elastic, how long it is until the ungee jumper returns to the lowest position? 7 cclist rides one lap of a circular track at a constant speed so that her distance, d metres, from her starting point at an time, t seconds, after starting is: t d = 0 0 cos a the time taken to complete one lap the radius of the track c the maimum distance from the start d the length of the track e the distance from her starting point after i seconds and ii 0 seconds f the times at which she is 9. metres from her starting point. 8 The depth of water, d metres, at a port entrance is given the function t d(t) =. +. sin where t is in hours. a Find i the maimum and ii the minimum depth at the port entrance. certain ship needs the depth at the port entrance to e more than metres. The ship can e loaded and unloaded, and in and out of the port, in 9 hours. ssuming that the ship enters the port just as the depth at the entrance passes metres, will the ship e ale to eit 9 hours later? How long will it have to spare, or how man minutes will it miss out?

51 Chapter Circular functions summar Trigonometric ratios SOH CH TO O sin = --- cos = --- tan = H H ngles of elevation and depression O --- H O ngle of depression ngle of elevation Horizontal The unit circle sin = cos = nd quadrant, 80 S T 90 cos C st quadrant P( ) sin 0, 0, rd quadrant th quadrant 70 tan = height of point where angle line meets tangent line at RHS of unit circle tan = -- tan

52 Mathematical Methods Units and For eact values use special triangles: 0 0 Radians c = 80 c 80 = ---- = -- c 80 Eact values For eact values use special triangles: 0 0 ngle ( c ) sin cos tan or 0 -- = -- or = = -- or or 90 0 undefined

53 Chapter Circular functions 7 Smmetr unit circle sketch of the given angle related ack to the first quadrant is often helpful. Provided an angle is epressed as ± or ±, the trig function (sin or cos) remains the same, onl the sign (+ or ) ma change. So ma e written as , -----, -----, etc. 8 sin ( ) = sin cos ( ) = cos tan ( ) = tan sin ( + ) = sin cos ( + ) = cos tan ( + ) = tan sin ( ) = sin cos ( ) = cos tan ( ) = tan Identities sin + cos = sin tan = cos sin(90 ) = cos or sin -- cos(90 ) = sin or cos -- = cos = sin Sine and cosine graphs Basic graph tpes:. = a sin = a cos. Period = -----, amplitude = a a = a sin a = a cos a 0 = period a 0 Sketching:. Recall asic graph tpe.. Find period and amplitude.. Sketch in sections of one period.. Find intercepts etween other known intercepts or minimum and maimum points.

54 8 Mathematical Methods Units and Tangent graphs = a tan Period = -----, no amplitude smptotes at = ± -----, and ever period to the left and right of these. The formula for asmptotes applies: ( n + ) asmptote = ± -----, where n = 0,,,... 0 Solving trigonometric equations When solving trigonometric equations, the following need to e determined:. The first quadrant angle, irrespective of the sign.. The two quadrants in which the given function is positive or negative.. Two solutions etween = 0 and = (use the appropriate sine, cosine or tangent smmetr propert). If more solutions are required:. Repeatedl add (or sutract) the period to the two solutions as man times as required, noting solutions after each addition or sutraction.. Stop when all solutions within the specified domain are found. pplications of sine and cosine functions To find the maimum value of a function, replace sin or cos with + To find the minimum value of a function, replace sin or cos with Initial values occur at t = 0 sketch graph ma provide greater understanding.

55 Chapter Circular functions 9 CHPTER review Multiple choice The relationship etween a, and in the triangle is: tan a = -- B sin a = -- C tan a = -- D cos a = E sin a = -- a In the triangle, the value of is closest to:. B.8 C.07 D 7. E.8. The angle that the -metre ladder makes with the wall in this diagram is closest to:. B.8 C..0 m D 8 E 7. m In which quadrants is tan positive? and B and C and D and E and The value of cos 0 is: positive, as 0 is in the st quadrant B negative, as 0 is in the nd quadrant C negative, as 0 is in the rd quadrant D negative, as 0 is in the th quadrant E positive, as 0 is in the th quadrant. If tan a < 0, sin a < 0 and 0 < a < 0, then which one of the following is correct? 0 < a < 90 B 80 < a < 70 C 90 < a < 80 D 0 < a < 80 E 70 < a < 0 7 If tan =, then sin could e equal to: -- B C D E 0 B B B B

56 0 Mathematical Methods Units and C C D 8 The angle 0 is equivalent to: B C D E The angle which is equivalent to is: 0 B C 0 D 70 E 00 0 The value of cos is: B -- C D E D E E E E Use the following information to answer questions to : sin a = 0. and 0 < a < --. sin ( a) is equal to: 0. B 0.8 C 0. D 0.8 E cos a is equal to: 0. B 0.8 C 0. D 0.8 E 0 tan ( a) is equal to: 0.7 B. C. D 0.7 E. If sin = 0.9 and -- < <, then tan is equal to: 0.97 B 0.97 C 0.0 D 0.97 E 0.0 The one value which is equal to sin is: tan 7 B cos 7 C cos D sin 7 E cos 7 Questions to 8 refer to the function: f() = sin. F F F F The amplitude of f() is equal to: B C D E 7 The period of f() is equal to: B C D E 8 The range of f() is: [0, ] B [0, ] C [, 0] D [, ] E [, ] 9 The rule for this graph is: = cos -- C = cos -- E = cos B = sin D = cos 0

57 Chapter Circular functions 0 The function = tan has a period and asmptote respectivel of: and = B and = -- C -- and = -- 8 D and = 8 E -- and = -- If sin = 0.9, then could e equal to: B 7 C 0 D E 0 If tan = 0.89 has one solution of = 0, then another solution could e: 0 B 0 C 00 D 80 E 70 The maimum value of = sin -- is: B 0 C D E The minimum value of h = cos t + is: B C 8 D E G H H I I Short answer The angle of elevation from an oserver to an aircraft when it is. km awa is. How high is the aircraft aove the ground if the oserver s ee level is.7 m aove the ground? (Give our answer to the nearest metre.).7 m. km Find the eact values of: a tan 0 cos 0 c sin If cos = -- and 0 < < --, find the eact values of: a sin tan Convert the following angles to eact radians. a 0 0 c Convert the following radian measures to the nearest tenth of a degree. 7 a c If cos = 0.9, find: a sin (80 ) cos ( ) c tan ( + ) 7 If cos = 0.9 and -- < <, evaluate: a sin tan 8 Sketch the graphs of the following functions. a = sin -- over [, ] =. cos over [, ] B B C C D E F

58 Mathematical Methods Units and G H H 9 Sketch the graph of: a = tan for [, ] = -- tan -- for [0, ]. 0 Find all of the solutions to the equation sin = over the domain [0, ]. Solve the equation + cos = 0 over the domain [0, 0 ]. nalsis standing wave on a guitar string ma e approimated the function = 0. sin -----, L 0 where cm and cm are defined on the diagram at right. a Find the period of the standing wave. If the frets coincide with the mean positions of the wave, find the value of L. c If the frets were to e spaced at cm, what would e the equation of a similar standing wave of amplitude 0. cm such that a fret is at each mean position? The numer of raits in a national park is oserved for one ear. t an time, t months after oservation egins, the numer is modelled the function: t P = 0.8 sin ---- where P is in thousands. a Find: i the maimum numer of raits ii the minimum numer of raits iii the median numer of raits. Find i the period and ii the amplitude of the function. c Sketch the graph of the function. d Find the population after months. e How long is the population elow 00? f How long is the population aove 00? test ourself CHPTER