Chapter 1. The word trigonometry comes from two Greek words, trigonon, meaning triangle, and. Trigonometric Ideas COPYRIGHTED MATERIAL

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1 Chapter Trigonometric Ideas The word trigonometr comes from two Greek words, trigonon, meaning triangle, and metria, meaning measurement This is the branch of mathematics that deals with the ratios between the sides of right triangles with reference to either of its acute angles and enables ou to use this information to find unknown sides or angles of an triangle Trigonometr is not just an intellectual eercise, but has uses in the fields of engineering, surveing, navigation, architecture, and es, even rocket science Angles and Quadrants An angle is a measure of rotation and is epressed in degrees or radians For now, we ll stick with degrees, and we ll eamine working with radians in the net chapter Consider an angle in standard position to have its verte at the origin (the place where the - and -aes cross), labeled O in the diagrams Angle measure is the amount of rotation between the two ras forming the angle II III O 45 COPYRIGHTED MATERIAL I IV P A first quadrant angle in standard position Figure the initial side of the angle above beginning on the -ais to the right of the origin Consider the terminal side of the angle to be hinged at O The terminal side of the angle, OP, was rotated counterclockwise from the -ais through an angle of less than 90 to form the first quadrant angle shown above Notice the Roman numerals The mark the quadrants I, or first; II, or second; III, or third; and IV, or fourth Notice that the quadrant numbers rotate counterclockwise around the origin Because the angle in the above figure has its initial side on the -ais, it is said to be in standard position Had the terminal side made a full turn and come back to the -ais, it would have rotated 60

2 CliffsStudSolver Trigonometr Q II I O 5 III IV A second quadrant angle in standard position The above figure is called a second quadrant angle because its terminal side is in the second quadrant When the magnitude of an angle is measured in a counterclockwise direction, the angle s measure is positive The above figure shows an angle of 5 measure R II I -5 O III IV A second quadrant angle measured clockwise The angle in the above figure is identical to the figure that precedes it in ever wa ecept how the angle was measured Since it was measured clockwise rather than counterclockwise, it has a measure of -5 Notice that the absolute value of that angle is obtained b subtracting 5 from 60 The negative sign marks the direction in which it was measured

3 Chapter : Trigonometric Ideas II I O -0 III IV S A fourth quadrant negative angle Notice that the fourth quadrant angle in the above figure, if measured counterclockwise, would have measured 0 Can ou see wh? Moving counterclockwise, it would have been 0 sh of a full 60 rotation II I 70 O III IV A quadrantal angle When an angle is in standard position, and its terminal side coincides with one of the aes, it is referred to as a quadrantal angle Angles of 90, 80, and 70 are three eamples of quadrantal angles The are b no means all the quadrantal angles that are possible, but we ll get to that in the net lesson

4 4 CliffsStudSolver Trigonometr II I 40 III IV A third quadrant angle The one angle remaining to be shown is a Q-III angle (see above) A third quadrant (or Q-III) angle is an angle with its terminal side being in the third quadrant Because this angle was formed b a counterclockwise rotation, it is positive Eample Problems These problems show the answers and solutions In which quadrant is the terminal side of a 95 angle in standard position? answer: II This also breaks with the stle from CliffsStudSolver Geometr and CliffsStudSolver Algebra Since the angle is in standard position, its initial side is on the -ais to the right of the origin The -ais forms a right (90 ) angle with the initial side, so a 95 angle s terminal side must sweep past the vertical -ais and into quadrant II In which quadrant is the terminal side of a -0 angle? answer: I Since the angle is in standard position, its initial side is on the -ais to the right of the origin Since its sign is negative, its terminal side rotates clockwise past the -ais at -90, on past the -ais at -80, past the -ais again at -70, and continues on another 50 to terminate in the first quadrant See the figure that follows

5 Chapter : Trigonometric Ideas 5 II I -00 III IV A -0 angle What is the special name given to a right angle in standard position? answer: quadrantal angle A right angle in standard position will have its terminal side on the -ais That makes it a quadrantal angle Coterminal Angles Two angles that are in standard position and share a common terminal side are said to be coterminal angles All of the angles in the following figure are coterminal with an angle of degree measure of 45 The arrow shows the direction and the number of rotations through which the terminal side goes Angles coterminal with 45

6 6 CliffsStudSolver Trigonometr All angles that are coterminal with an angle measuring d ma be represented b the following equation: d + n 60 Eample Problems These problems show the answers and solutions Name four angles that are coterminal with 80 answer: -640, -80, 440, 800, An angle coterminal with 80 must have a multiple of 60 added to it; according to the formula d + n 60, the following are some angles coterminal with 80 : d + n 60 = 80 + ()(60 ) = = 440 d + n 60 = 80 + ()(60 ) = = 800 d + n 60 = 80 + ()(60 ) = = 60 d + n 60 = 80 + (4)(60 ) = = 50 Of course, an number could have been substituted for n, and that means negative as well as positive values, for eample: d + n 60 = 80 + (-)(60 ) = = -80 d + n 60 = 80 + (-)(60 ) = = -640 d + n 60 = 80 + (-)(60 ) = = -000 d + n 60 = 80 + (-4)(60 ) = = -60 So, the answer is reall the following: Four angles coterminal with 80 are, -640, -80, 80, 440, 800, The angle 80 itself was included in the series so as not to break the pattern Notice that as ou move from left to right, each angle measure is 60 greater than the one to its left Is an angle measuring 0 coterminal with an angle measuring 960? answer: No If angles measuring 0 and 960 were coterminal, then 960 = 0 + n = n 60 But 740 is not a multiple of 60, so the angles cannot be coterminal

7 Chapter : Trigonometric Ideas 7 Work Problems Use these problems to give ourself additional practice In which quadrant does the terminal side of an angle of degree measure 00 fall? In order for an angle to be a quadrantal angle, b what number must it be capable of being divided? What is the lowest possible positive degree measure for an angle that is coterminal with an angle of -770? 4 In which quadrant does the terminal side of an angle of 990 degree measure fall? 5 Name two positive and two negative angles that are coterminal with an angle of degree measure 5 Worked Solutions II First see how man times 60 can be subtracted or divided out of the total 60 = = 0 A 0 angle is larger than 90 and less than 80, and so its terminal side falls in the second quadrant 90 Quadrantal angles terminal sides fall on the aes Therefore, no matter what the size of the coterminal angle, it must be capable of being in the positions of 90, 80, 70, or 0 All are divisible b 90 0 Just keep adding 60 until the sum goes from a negative to a positive value: 5 60 = = 0 4 None It is quadrantal 990 is two full revolutions from the starting position, and then an additional 70 : 60 = = 70 That places the terminal side on the -ais, south of the origin (negative) 5 585, 5, 495, 855 Needless to sa, there are an infinite number of other solutions, each of which is determined b substituting into the epression: d + n 60

8 8 CliffsStudSolver Trigonometr Trigonometric Functions of Acute Angles The building blocks of trigonometr are based on the characteristics of similar triangles that were first formulated b Euclid He discovered that if two triangles have two angles of equal measure, then the triangles are similar In similar triangles, the ratios of the corresponding sides of one to the other are all equal Since all right triangles contain a 90 angle, proving two of them similar onl requires having one acute angle of one triangle equal in measure to one acute angle of the second Having established that, we easil find that in two similar right triangles, the ratio of each side to another in one triangle is equal to the ratio between the two corresponding sides of the other triangle It is no long stretch from there to realize that this must be true of all similar triangles Those relationships led to the trigonometric ratios It is customar to use lowercase Greek letters to designate the angle measure of specific angles It doesn t matter which Greek letter is used, but the most common are α (alpha), β (beta), φ (phi), and θ (theta) The trigonometric ratios that follow are based upon the following reference triangle, which is drawn in two different was c a hpotenuse opposite side b (a) adjacent side (b) Figures (a) and (b) Both figures show the same triangle with sides a, b, and c, and with angle θ at the left end of the base The difference is that in figure (b), the two legs are labeled with respect to θ That is to sa, side a is marked as opposite to θ, and side b is adjacent to θ You might correctl argue that side c is also adjacent to θ, but that side alread has a name (ou learned about this in plane geometr) Being the side opposite the right angle, it s the hpotenuse, hence it is the nonhpotenuse adjacent side to θ that is assigned the name adjacent That leads us to the first three trigonometric functions: length of side opposite The sine of θ is: sin c a i i = = length of hpotenuse The cosine of θ is: cos b length of side adjacent i i = c = length of hpotenuse length of side opposite The tangent of θ is: tan b a i i = = length of side adjacent i In the earl das of American histor, wa before the das of political correctness, some ambitious trigonometr student in search of a mnemonic device b which to remember his or her trigonometric ratios dreamed up the SOHCAHTOA Indian tribe, which toda would be the SOHCAHTOA tribe of Native Americans SOHCAHTOA is an acronm for the basic trig ratios and their components; that is: Sin-Opposite/Hpotenuse-Cos-Adjacent/Hpotenuse-Tan-Opposite/Adjacent

9 Chapter : Trigonometric Ideas 9 Keep in mind that as long as the angles remain the same, the ratios of their pairs of sides will remain the same, regardless of how big or small the are in length The trigonometric ratios in right triangles depend eclusivel on the angle measurements of the triangles and have no dependence on the lengths of their sides Eample Problems These problems show the answers and solutions All refer to the following figure β 6 cm cm α γ Find the sine of α answer: 05 opposite The sin ratio is hpotenuse cm is the length of the side opposite α, and the hpotenuse is 6 cm So, sina = 6 = = 05 Find cosβ answer: 05 adjacent The cos ratio is hpotenuse cm is the length of the side adjacent β, and the hpotenuse is 6 cm So, cosb = 6 = = 05 Find tanβ answer: opposite The tan ratio is adjacent cm is the length of the side opposite β, and its adjacent side is cm So, tanb = = Reciprocal Trigonometric Functions The three remaining trigonometric ratios are the reciprocals of the first three You ma think of them as the first three turned upside down, or what ou must multipl the first three b in order to get a product of The reciprocal of the sine is cosecant, abbreviated csc Secant is the reciprocal of cosine and is abbreviated sec Finall, cotangent, abbreviated cot, is the reciprocal of tangent

10 0 CliffsStudSolver Trigonometr c a hpotenuse opposite side θ b θ adjacent side (a) (b) Models for reciprocal trigonometric ratios length of hpotenuse The cosecant of θ is: csc i = c a = length of side opposite i length of hpotenuse The secant of θ is: sec i = c b = length of side adjacent i The cotangent of θ is: cot b length of side adjacent i i = a = length of side opposite i There is no SOHCAHTOA tribe here to help out, but ou shouldn t need one Just remember the pairings, find the right combination for its reciprocal (that is for secant; remember it pairs with cosine), and flip it over Eample Problems These problems show the answers and solutions All problems refer to the following figure φ 6 in 0 in 8 in θ Model for eample problems Find cscφ answer: 5 length of hpotenuse csc z = = 0 length of side opposite 8 = 5 z Find secθ answer: 5 length of hpotenuse sec i = = 0 length of side adjacent 8 = 5 i

11 Chapter : Trigonometric Ideas Find cotθ answer: length of side adjacent i cot i = = 8 length of side opposite 6 = i Work Problems Use these problems to give ourself additional practice An angle has a sine of 0 a What other function s value can ou determine? b What is that value? An angle has a cosine of 06 a What other function s value can ou determine? b What is that value? An angle has a tangent of 5 a What other function s value can ou determine? b What is that value? 4 An angle has a secant of 8 a What other function s value can ou determine? b What is that value? 5 An angle has a cosecant of 5 a What other function s value can ou determine? b What is that value? Worked Solutions cosecant, Sine s reciprocal function is cosecant It is the reciprocal of sine; that is, csc i = sini, but ou know that the value of sin is 0, therefore csc i = 0 Divide b 0, and ou get (rounding to the hundredths) secant, 67 Cosine s reciprocal function is secant Since it s the reciprocal of cosine, seci = cosi, but ou were given the cos as 06, therefore seci = 06 Divide b 06 to find a rounded value of 67 cotangent, 04 Tangent s reciprocal function is cotangent It is the reciprocal of tangent, so coti = tani, but ou were given tan = 5, so coti = 5 Divide b 5 and get 04

12 CliffsStudSolver Trigonometr 4 cosine, 056 Since secant s reciprocal function is cosine, cosi =, but ou know seci that the value of sec is 8; therefore cosi = 8 Divide b 8, and ou get 056 (rounding to the hundredths) 5 sine, 09 Since cosecant s reciprocal function is sine, sini =, but ou know that csc i the value of csc is 5; therefore sini = 5 Divide b 5, and ou ll get 088, which ou ll round to 09 Introducing Trigonometric Identities When trigonometric functions of an angleφ are related in an equation, and that equation is true for all values of φ, then the equation is known as a trigonometric identit The following trigonometric identities can be constructed from the trigonometric ratios that ou just reviewed c a φ b Triangle referenced b identities below Referring to the above figure, since sinz = c a, cosz = c b, and tanz = b a, it follows that sinz tan c a z = = cosz b c The second part of which ou can simplif like this: a c a z c $ b = $ b = b a z sinz Which serves to prove the identit: tanz = cosz cosz You could also prove the identit: cotz = sinz These two identities are etremel useful and should be memorized There s a third ver hand identit, but first, ou must become familiar with some conventional notation The smbol (sinθ) and sin θ mean the same thing and ma be used interchangeabl With that in mind, the third identit referred to is sin φ + cos φ = If ou would like to see that proven, refer to the previous figure and the Pthagorean theorem, stated as a + b = c sin cos c a c b z+ z= + = a b l b l + c b c You can add the two fractions to get a + b c

13 Chapter : Trigonometric Ideas But ou alread know that a + b = c, so ou substitute in the numerator and simplif: Therefore, sin φ + cos φ = The importance of these three identities cannot be over-stressed You will deal much more etensivel with identities in the fourth chapter, but for now, tr to learn these three c c = Eample Problems These problems show the answers and solutions Where applicable, round each answer to the nearest thousandth Find the sin and tangent of λ if λ is an acute angle (0 < λ < 90 ) and cosλ = cosm = 5 answer: 0980, 49 Since sin φ + cos φ =, then sin λ + cos λ = Substitute: sin m + b 5 l = Square and subtract from both sides: sin m = - 5 Subtract: sin m = 4 5 Take the square root of both sides: sin m = 4 5 = Net, find the tangent using what ou just found: cosλ = tanm = sinm and cosm cosm = 5 = 0 Substituting, ou find that tan m = = 49 Find the cos and tangent of φ if φ is an acute angle (0 < φ < 90 ) and sinφ = 0867 answer: 0498, 74 First, use the identit sin φ + cos φ = to find cosφ Substitute: (0867) + cos φ = Subtract (0867) from both sides: cos φ = - (0867) Square the quantit in parentheses: cos φ = Subtract 075 from : cos φ = 048 Take the square root of both sides: cosφ = 0498 Now, solve for the tangent

14 4 CliffsStudSolver Trigonometr First write the relevant identit: Substitute for sin and cos: sinz tanz = cosz tanz = And divide: tanz = 0 74 Find the sin of θ if θ is an acute angle (0 < θ < 90 ), tanθ = 9, and cosφ = 064 answer: 0766 This time, ou onl need: tan i = sini cos i Substitute what ou know: 9 = sin 0 64 i Now multipl both sides b 064: (064)(9) = sini b 0 64 l^ 0 64 h Reversing the results, ou get: sinθ = 0766 Trigonometric Cofunctions Trigonometric functions are often considered in pairs, known as cofunctions Sine and cosine are cofunctions So are secant and cosecant The final pair of cofunctions are tangent and cotangent From the right triangle ABC, the following identities can be seen: sina = a c = cosb sinb= b c = cosa seca = c b = csc B secb= c a = csc A tana = a b = cotb tanb= b a = cota B a c C b θ A Reference triangle for cofunctions To refresh our memor, all three angles of a triangle are supplementar (sum to 80 ), and angles that sum to 90 are known as complementar Since one of the three angles in a right triangle measures 90, the sum of the remaining acute angles must be complementar Refer to the above reference triangle to confirm the following relationships: sinθ = cos (90 -θ) secθ = csc (90 -θ) tanθ = cot (90 -θ) cosθ = sin (90 -θ) csc θ = sec (90 -θ) cot θ = tan (90 -θ)

15 Chapter : Trigonometric Ideas 5 Two Special Triangles The figure below shows an isosceles right triangle with each leg having a length of Can ou figure out the measure of angle X? 45 Y An isosceles right triangle That last question was intended as a joke, since in an isosceles triangle, the angles opposite the equal legs are alwas equal in measure What is different, however, is that in an isosceles right triangle, the acute angles alwas measure 45 That s because the must be both complementar and equal What is also different in an isosceles right triangle is that the hpotenuse alwas has the same relationship to the legs That relationship, of course, ma be found using the Pthagorean theorem In this case: z = + Substituting: z = + Squaring and adding: z = Therefore: z = All right, ou alread knew that, because it shows that in the above figure, but what if instead of, and had been? Substituting: z = + Squaring and adding: z = 8 Therefore: z = 8 = 4 = $ Do ou see the pattern et? Tr one more, just to make sure This time, let and be 5 Substituting: z = Squaring and adding: z = 50 Therefore: z = 50 = 5 = 5 X $ To sum it all up, when dealing with an isosceles right triangle, the hpotenuse is alwas the length of the leg times the square root of Z

16 6 CliffsStudSolver Trigonometr 60 S Q 0 The right triangle R Students and teachers of trigonometr are quite fond of a second special right triangle That s the one with acute angles of 0 and 60, or as it s often referred to, the right triangle The relationship among the sides are spelled out in the above figure Notice that the side opposite the 0 angle is half the length of the hpotenuse, and the length of the side opposite the 60 angle is half the hpotenuse times the square root of three From the two special triangles, ou can compile a table of frequentl used trigonometric functions Table of Trigonometric Ratios for 0, 45, and 60 Angles i sini cosi seci csci tani coti Eample Problems These problems show the answers and solutions One leg of an isosceles right triangle is 4 cm long a How long is the other leg? b How long is the hpotenuse? answer: 4 cm, 4 For part a, both legs of an isosceles right triangle are the same length As for part b, ou have seen that the hpotenuse of an isosceles right triangle is equal in length to a side times the square root of two The shortest leg of a triangle is 5 inches long a How long is the other leg? b How long is the hpotenuse? answer: 5, 0 in

17 Chapter : Trigonometric Ideas 7 It might be simpler to find the answer to part b first In a triangle, the shortest side is the one opposite the smallest angle, that is, the one opposite 0 That side is half the length of the hpotenuse, so the hpotenuse must be twice the length of that side, or 0 in Finall, the side opposite the 60 angle (the other leg) is half the hpotenuse times the square root of three In triangle ABC, with right angle at B, the cosine of A is What is the sine C? 5 answer: 5 The cofunction identities tell us that in a given right triangle, the sine of one acute angle is the cosine of the other Work Problems Use these problems to give ourself additional practice In triangle ABC, with right angle at B, the cosine of A is 5 What are the sine of A and the tangent of A? One leg of an isosceles right triangle is 8 cm long What is the length of the hpotenuse? The hpotenuse of a triangle is 0 inches long The shorter leg of the triangle is a, and the longer b Find the lengths of a and b 4 In triangle PQR, with right angle at P, the sine of Q is 05 What is the tangent of R? Worked Solutions 5 4, 4 Remember: sin A+ cos A= Therefore: sin A 5 + b l = Square and subtract from both sides: sin A = Make an equivalent fraction: sin A = - Subtract: sin A = 6 5 Now get the square root of both sides: sina = 4 5 As for tangent, use the ratio identit: tana = sina cosa 4 Substitute: tana = 5 5 Rewrite as a reciprocal multiplication: tana = # The 5s cancel, so ou get: tana 4 5 = # 4 5 =

18 8 CliffsStudSolver Trigonometr 8 cm You learned in the Two Special Triangles section that the hpotenuse of an isosceles right triangle is equal to the length of a side times the square root of two If ou did not recall that, then use the Pthagorean theorem, which in the case of an isosceles right triangle ma be written c = a + a (Remember, both legs are equal) Substitute: c = Square and add: c = = 8 Solve for c: c = 8 5 inches, 5 inches In a triangle, the shorter leg is opposite the 0 angle and is half the length of the hpotenuse Half of 0 inches is 5 inches The longer leg is opposite the 60 angle and is equal to half the hpotenuse times the square root of That s 5 inches The sine of Q is 05, but ou want the tangent of R, so ou ll use sine s cofunction, cos R= 05 Net, ou need to find sin R so that ou ma relate sin and cos with the tangent identit First write the equation: sin R + cos R = Net, substitute: sin R + (05) = Clear the parentheses: sin R = Collect the constants: sin R = Subtract: sin R = 0945 Solve for sin R: sin R = 097 Now for the tangent identit: Substitute: tanr = sinr cosr tanr = And divide: tanr = 7 67 Functions of General Angles When an acute angle is written in standard position, it is alwas in the first quadrant, and there, all trigonometric functions eist and are positive This is not true, however, of angles in general Some of the si trigonometric functions are undefined for quadrantal angles, and some have negative values in certain quadrants In standard position, an angle is considered to have its starting position in quadrant I on the -ais and its terminal side in or between one of the four quadrants Consider the angle, φ in the following figure Point P is on the terminal side of the angle, r, and has coordinates (, ) The radius of the circle is

19 Chapter : Trigonometric Ideas 9 P(,) r φ A unit circle (r = ) Look it over carefull, and ou ll see that sinφ =, cosφ =, and tanz = sinz This is a tangible proof of the tanz = ratio identit cosz Alas, not all standard angles terminate in the first quadrant Look at these three angles: P(-,) r φ 0 r P(-,-) φ 0 φ 0 r P(,-) (a) Angles terminating in quadrants other than I: (a) Angle terminating in II; (b) Angle terminating in III; (c) Angle terminating in IV In all four quadrants, the value of r is positive In quadrant II, onl sine is positive d r n, while cosine b - r l and tangent d- n are negative Cosecant, secant, and tangent will alwas have the same signs as their inverses We leave it to ou to figure out what functions are positive or negative in quadrants III and IV, and wh To help in the future, there s a little mnemonic scheme ou might want to remember, and it s represented in the following figure (b) (c)

20 40 CliffsStudSolver Trigonometr S A II I III 0 IV T C Clockwise ACTS, or counter CAST You ma remember this as being the word CAST reading counterclockwise and beginning in quadrant IV, or ACTS reading clockwise from quadrant I The letters tell ou which ratio (and its inverse) is positive in that quadrant: C for cosine in IV, T for tangent in III, S for sine in II, and A for all in I Should z be a quadrantal angle, then either or ma be equal to 0 If that 0 is in the numerator, then the trigonometric ratio will have a value of 0, but if it s in the denominator, the ratio is undefined Yet other times, the ratio equals That s shown in the following table Trigonometric Ratios for Quadrantal Angles i sini cosi tani seci csci coti Eample Problems These problems show the answers and solutions What is the sign of tan 0? answer: positive Refer to the previous figure 0 falls in quadrant III, where tangent is positive What is the sign of sin 00? answer: negative Refer to the previous figure 00 falls in quadrant IV, where sine is negative

21 Chapter : Trigonometric Ideas 4 What is the sign of sec 0? answer: positive Secant is the reciprocal of cosine 0 falls in quadrant IV, where cosine is positive, so secant must also be positive in that quadrant Reference Angles The trigonometric functions of nonacute angles ma be converted so that the correspond to the functions of acute angles See the following figure P(-,) r β α 0 80 α = reference β a is a second quadrant angle in standard position B subtracting a from 80, ou get the acute reference angle, b You can now find the trigonometric ratios using the reference angle, but bear in mind that since the original angle terminated in the second quadrant, all of the ratios will be negative ecept for sine and cosecant sinb = r cosb = - r tanb = - To find cosecant, secant, and cotangent, just flip over the three preceding ratios α r P(-,-) β 0 α 80 = reference β

22 4 CliffsStudSolver Trigonometr In the third quadrant, reference β is found b subtracting 80 from α, as shown in the previous figure sinb = - r cosb = - r - tanb = - α 0 β r P(,-) 60 α = reference β In quadrant IV, β is found b subtracting α from 60 sinb = - r cosb = r tanb = - Notice that the reference angle alwas sits above or below the -ais Eample Problems These problems show the answers and solutions What is the cosine of 50? answer: 0867 or - To find the reference angle, subtract 50 from 80 That s a 0 angle In a triangle of hpotenuse, the side opposite the 0 angle is, and the side opposite the 60 angle is That makes cos0c =, but cosine is negative in the second quadrant 7, hence the two possible answers What is the sine of 5? answer: 0707 or - To find a quadrant III reference angle, subtract = 45 When ou studied an isosceles right triangle, ou found both sin and cos45c =, which ou rationalize b

23 Chapter : Trigonometric Ideas 4 multipling it b, thus getting It s considered poor form to leave a radical in the denominator 44, so half of it is 0707, but since sine is negative in the third quadrant, the solution is negative Find the tangent of 00 answer: 7 or - The angle is in quadrant IV, so subtract it from 60 60c- 00c= 60c Refer back to solution, to find that the tan60c = = In quadrant IV, tan is negative, hence the answers shown Squiggl versus Straight We presume that ou noticed the squiggl equal signs used in the eplanations of eample problems and Mathematicians use that smbol to indicate that one quantit is a close approimation of another For eample, is easil shown on a calculator to equal , and then some, so mathematicians would much prefer to write 7 than = 7 On the other hand, a mathematician would never write In fact, =, with no approimating or rounding involved The squiggl equal sign is also likel to be used when the numbers are readings from scientific instruments, which onl approimate true conditions Work Problems Use these problems to give ourself additional practice A 5 angle is angle in standard position What is its cosecant? Find the cotangent of a 40 angle in standard position What is the sine of a 675 angle in standard position? 4 Find the secant of a 0 angle in standard position 5 Angle χ is in standard position, and its terminal side passes through the point with coordinates (-,5) Find all si of its trigonometric functions 6 If sini = 5 4, and cosθ is negative, what are the values of the five remaining trigonometric functions?

24 44 CliffsStudSolver Trigonometr Worked Solutions or 44 The reference angle for 5 is 45 If necessar, refer back to the figure on p 5 ( An isosceles right triangle ), where ou ll find that csc 45 is, which, of course, is simpl Since sine is positive in the second quadrant, so is cosecant or 0577 To find the reference angle, subtract 80 That makes 60 Net, if necessar, look back at the figure on p 6 ( The right triangle ) cot60c =, which ou rationalize: $ = Finall, consider that the angle originall terminated in Q-III (third quadrant), where tangent, and therefore, cotangent, is positive - or goes one full 60 turn, and then 45 less than another one That makes it a 45 angle in Q-IV, just below the -ais sin 45, as ou ma recall from the figure on p 5 ( An isosceles right triangle ) or the table on p 6 (Table of Trigonometric Ratios for 0, 45, and 60 Angles), = Sine is not positive in Q-IV, so the answer will be negative 4 or 55 Subtract from 60, and ou find that this is a 0 angle in Q-IV Secant 0 is Since cosine is positive in Q-IV, so is secant 5 Since the angle s terminal side passes through (-, 5), build a right triangle about that point in Q-II, with the angle to be dealt with at the origin The side opposite that angle is the -coordinate, 5 The adjacent side of the triangle is the absolute value of the -coordinate, - = To find the hpotenuse of the triangle, ou can either use the Pthagorean theorem, or remember the Pthagorean triple, 5-- Either wa, the hpotenuse has a length of So, take care to remember that the terminal side of the angle is in Q-II, where onl sine and its reciprocal, cosecant, are positive With that in mind, ou find: sin =, cos, tan, csc, sec, cot 5 =- =- 5 = 5 =- and =- 5 6 Since sini = 5 4 and cosθ is negative, the angle must be in the second quadrant; since in Q-I, cosine would have been positive, and sine is positive in Q-III and Q-IV The hpotenuse of the triangle must be 5, and the opposite side 4 (from the sine), which makes the adjacent side, either b the Pthagorean theorem, or b remembering the Pthagorean triple, -4-5 So, cos i =- 5 and tani=- 4, csc i= 5 4, seci=- 5, and cot i =- 4 Trig Tables versus Calculators We ve alread taken note of the fact that the trigonometric ratios depend onl on the value of the angle measure and are independent of the size of the triangle s sides You would think that somebod would write down those values for each degree of measure It should come as no surprise then, that somebod has You ll find a table of trigonometric ratios in the back of this book, beginning on p 97 In order to read the value of a trigonometric ratio for a certain angle,

25 Chapter : Trigonometric Ideas 45 ou move one finger down the column that sas the name of the ratio ou re looking for at the top, while moving our ees down the angle values on the left until ou find the one for which ou re looking Then move them across to the proper column Where our ees and finger meet should be the value ou want The tables show values for all si trigonometric functions in increments of Now all of the angles we ve spoken about so far have been in degrees, but the degree is not the smallest unit of angle measure Just like that other round thing we re used to seeing on a wall, degrees can be broken down into minutes There are, in fact, 60 minutes in one degree, and, in turn, 60 seconds in one minute What that means is that it is possible to break the degree measurement of a circle into =,96,000 parts All of those parts are not especiall useful or accurate when dealing with the small circles in this book or on our graph paper, but consider how hand the can be when describing the location on a three-dimensional circular object, like the earth In addition to the traditional subdivisions mentioned in the last paragraph, degrees are also capable of being subdivided into decimal parts, like 5, 4075, or even 765 There is reall no limit to the number of decimal places to which an angle s size ma be carried out, although after the sith, one might start to become a bit suspicious about how accuratel that quantit was measured It is also true, however, that scientific measuring techniques are continuall being refined That brings us to scientific calculators Scientific calculators have the capacit to calculate at least the three fundamental trigonometric functions: sine, cosine, and tangent The method in which the data is entered varies from brand to brand, but for the most part, ou press the button that names the ratio ou want to find, enter the number of degrees and an decimal portion that ma follow, and press <Enter> to displa the ratio Most calculators do not provide cosecant, secant, or cotangent kes, however In order to find the cosecant of, sa 0, ou would first press the <SIN> button, tpe 0, and press <Enter> The screen now displas 05 Net, clear the screen or add the answer to memor and clear the screen Net tpe, or / depending upon how our calculator is labeled; the mean the same thing Then either press our <RECALL> ke or ke in 5 Finall, press <Enter> The screen will show, the cosecant of 0 That s because, as ou ll hopefull recall, cosecant and sine are inverse functions: csc f z = p sinz When using a table of trigonometric functions, it s possible to look at the function and see what the angle is Go ahead Look at the sines in the table on p 97 (Trig Functions Table) and find the angle whose sine is 04 Go ahead, reall do it We ll wait Well, if ou actuall went and looked under the sin column and found the value 04, ou would have found that it belongs to the angle, 5 You can t do the same thing with a calculator Instead, ou have to use the functions arcsin, arccos, and arctan, which are also written sin -, cos -, and tan - That means, the epression sinθ = 05 ma be written as sin - θ = 05 or arcsinθ = 05 Sometimes, it is referred to as the inverse sine of 05 In an of those cases, the epression is used if the trigonometric ratio is known, and the angle that it belongs to is being sought In the last case, when sin - θ = 05, then θ = 0 Calculators handle arcsine in different was On some, ou push an <arc> button and then press the appropriate trig ratio On others, ou must press the <nd function> button and then press the appropriate trig ratio button Interpolation Most trigonometric functions, whether listed in tables or found on calculators, are approimations Nevertheless, ever effort possible is made to get the numbers as close as possible to the actual value Bearing that in mind, if ou re using tables that give values to the nearest degree,

26 46 CliffsStudSolver Trigonometr and ou need the cosine of a fractional angle, sa 06, ou re going to need a wa to find a value that is more accurate than 08660, the cosine for 0 The method for approimating the closer value is known as interpolation It works like this: cos0 > cos06 > cos (which means cos06 is between cos0 and cos ) cos 0 = 08660, cos = 0857 The difference between them is = is 06, or 0 6 of the wa between the two angles, so cos 06 should be 06 of the difference between the two cosines less than cos 0 : = 0005 Since cos is lower than cos 0, subtract from cos 0 : = cos 06 = (The smbol means therefore ) Let s tr one more of those, so that ou have a chance to see both possible scenarios This time, find the sine of 4 4 < 4 < 4 sin 4 = 0669, sin 4 = 0680 The difference between them is = is 0, or 0 of the wa between the two angles, so sin 4 should be 0 of the difference between the two sines greater than sin 4 : = 0009 Since sin 4 is lower than sin 4 add to sin 4 : = 0670 sin 4 = 0670 Interpolation ma also be used to approimate the size of an angle to the nearest tenth of a degree Suppose that ou have sinθ = 09690, and ou know that sin 75 = and sin 76 = 0970 Z sin75c = _ ] ) 0 sinz = b [ `0 004^ = the differenceh ] sin76c = b \ a Now, we ll use the variable to set up a proportion: = = 000 = = 07 φ = = 757

27 Chapter : Trigonometric Ideas 47 Eample Problems These problems show the answers and solutions Use the tables that begin on p 97 Do not use a calculator How man seconds are there in a 5 angle? answer: 8,000 There are 60 minutes in one degree, and 60 seconds in each of those minutes That s for one degree: = 600 For 5 degrees, multipl it b 5: = 8,000 Find tan 468 to the nearest ten thousandth (four decimal places) answer: < 468 < 47 tan 46 = 055, tan 47 = 074 The difference between them is = is 08, or 8 of the wa between the two angles, so tan 468 should be 08 of the 0 difference between the two tans greater than tan 46 : = 0095 Since tan 46 is lower than tan 47 add to tan 46 : = 0650 tan 468 = 0650 For what acute angle does cosine have a value of 0786 if cos 8 = and cos 9 = 0777? answer: 85 Z cos75c = _ ] ) 0 cosz = b [ `0 009^ = the differenceh ] cos76c = b \ a Now, we ll use the variable to set up a proportion: = = = = 05 φ = = 85

28 48 CliffsStudSolver Trigonometr Work Problems Use these problems to give ourself additional practice How man seconds are there in a 0 angle? Find sin 57 to the nearest ten thousandth (four decimal places) Find tan 6 to the nearest ten thousandth (four decimal places) 4 What is the secant of a 765 angle in standard position? 5 For what acute angle does sine have a value of 0460 if sin 7 = and sin 8 = 04695? Worked Solutions 08, seconds per minute 60 minutes per hour 0 degrees gives = 08, < 57 < 6 sin 5 = 0576, sin 6 = The difference between them is = is 07, or 7 of the wa between the two angles, so sin 57 should be 07 of the 0 difference between the two sines greater than sin 5 : = Since sin 5 is lower than sin 6 add to sin 5 : = 0585 sin 57 = < 6 < 6 tan 6 = 8040, tan 6 = 8807 The difference between them is = is 0, or of the wa between the two angles, so sin 6 should be 0 of the 0 difference between the two tans greater than tan 6 : = 000 Since tan 6 is lower than tan 6 add to tan 6 : = 87 tan 6 = 87

29 Chapter : Trigonometric Ideas First of all, a 765 angle is a 45 angle after the two full rotations are removed Secant is the inverse of cosine, so find cos 45 = 707 Then make the fraction and divide 0707 = Alternatel, ou could have recalled that cos 45c =, so sec45c = You can rationalize the denominator b multipling to get $ =, = or 44 Z sin7c = _ ] * sin b [ i = `0 055^ = the differenceh ] sin8c = b \ a Now, ou ll use the variable to set up a proportion: = = 0006 = = 04 i = = 74 Chapter Problems and Solutions Problems Solve these problems for more practice appling the skills from this chapter Worked out solutions follow the problems In which quadrant is the terminal side of a 05 angle in standard position? What is the special name given to a straight angle in standard position? Is an angle measuring 45 coterminal with an angle measuring 975? 4 In order for an angle to be a quadrantal angle, b what number must it be capable of being divided? 5 In which quadrant does the terminal side of an angle of 60 degree measure fall?

30 50 CliffsStudSolver Trigonometr β 0 cm 5 cm α γ Problems 6-7 refer to the above figure 6 Find the sin of α 7 Find tanβ θ 8 in 0 in 6 in φ Problem 8 refers to the above figure 8 Find secθ 9 An angle has a sine of 04 What is the value of its cosecant? 0 An angle has a tangent of 5 What is the value of its cotangent? An angle has a cosecant of 8 What is the value of its sine? Find the cos and tangent of φ if φ is an acute angle (0 < φ < 90 ), and sinφ = One leg of an isosceles right triangle is 8 cm long a How long is the other leg? b How long is the hpotenuse? 4 In triangle ABC, with right angle at B, the cosine of A is 7 What is the sin C? 8 5 One leg of an isosceles right triangle is in long What is the length of the hpotenuse? 6 In triangle PQR, with right angle at P, the sine of Q is 05 What is the tangent of R?

31 Chapter : Trigonometric Ideas 5 7 What is the sign (positive or negative) of sin 90? 8 What is the cosine of 40? 9 Find the tangent of 0 0 Find the cotangent of a 40 angle in standard position Find the secant of a 00 angle in standard position If i = 5 4, and cosθ is negative, what are the values of the five remaining trigonometric functions? Use the tables that begin on p 97 Do not use a calculator Find tan 40 to the nearest ten thousandth (four decimal places) 4 Find sin 78 to the nearest ten thousandth (four decimal places) 5 For what acute angle does sine have a value of 0579 if sin 5 = 0576 and sin 6 = 05878? Answers and Solutions Answer: II Since the angle is in standard position, its initial side is on the -ais to the right of the origin the -ais forms a right (90 ) angle with the initial side, so a 05 angle s terminal side must sweep past the vertical and into quadrant II Answer: quadrantal A straight angle in standard position will have its terminal side on the -ais That makes it a quadrantal angle Answer: no If angles measuring 45 and 975 were coterminal, then 975 = 45 + n = n 60 But 70 is not a multiple of 60, so the angles cannot be coterminal 4 Answer: 90 A quadrantal angles terminal side falls on an ais Therefore, no matter what the size of the coterminal angle, it must be capable of being in the positions of 90, 80, 70, or 0 All are divisible b 90 5 Answer: None It is quadrantal 60 is three full revolutions from the starting position, and then an additional 80 : 60 = = 80 That places the terminal side on the -ais, left of the origin (negative)

32 5 CliffsStudSolver Trigonometr 6 Answer: 05 opposite The sin ratio is hpotenuse 5 cm is the length of the side opposite α, and the hpotenuse is 0 cm So, sina = 0 5 = = 05 7 Answer: opposite The tan ratio is adjacent 5 cm is the length of the side opposite β, and its adjacent side is 5 cm So, 5 tanb = 5 = 8 Answer: 5 length of hpotenuse sec i = = 0 length of side adjacent 8 = 5 i 9 Answer: 5 Cosecant is the reciprocal of sine; that is, csc i =, but ou know that sini the value of sin is 04; therefore csc i = 04 Divide b 04, and ou get 5 0 Answer: 04 Cotangent is the reciprocal function of tangent, so coti =, but ou tani were given tan = 5, so coti = 5 Divide b 5 and get 04 Answer: 06 Sine is cosecant s reciprocal function, sini =, but ou know that csc i the value of csc is 8; therefore, sini = 8 Divide b 8, and ou ll get , which ou ll round to 06 Answer: 0607, 04 First, use the identit sin φ + cos φ = to find cos φ Substitute: (0776) + cos φ = Subtract (0776) from both sides: cos φ = - (0776) Square the quantit in parentheses: cos φ = Subtract 060 from : cos φ = 0978 Take the square root of both sides: cos φ = 0607 Now, for the tangent First, write the relevant identit: Substitute for sin and cos: sinz tanz = cosz tanz = And divide: tanz = 04 Answer: 8 cm, 8 cm For part a, both legs of an isosceles right triangle are the same length As for part b, ou have seen that the hpotenuse of an isosceles right triangle is equal in length to a side times the square root of two, but if ou forgot that there s alwas the Pthagorean theorem

33 Chapter : Trigonometric Ideas 5 4 Answer: 8 7 The cofunction identities tell us that in a given right triangle, the sine of one acute angle is the cosine of the other 5 Answer: in In the section Two Special Triangles, the hpotenuse of an isosceles right triangle is equal to the length of a side times the square root of If ou did not recall that, then use the Pthagorean theorem, which in the case of an isosceles right triangle ma be written c = a + a (remember, both legs are equal) Substitute: c = + Square and add: c = = 88 Solve for c: c = 6 Answer: 409 The sine of Q is 084, but ou want the tangent of R, so ou ll use sine s cofunction, cos R = 084 Net, ou need to find sin R so that ou ma relate sin and cos with the tangent identit First write the equation: sin R + cos R = Net, substitute: sin R + (084) = Clear the parentheses: sin R = Collect the constants: sin R = Subtract: sin R = 0854 Solve for sinr sinr = 09 Now for the tangent identit: Substitute: tanr = sinr cosr tanr = And divide: tanr = Answer: negative 90 falls in quadrant IV, where sine is negative You ma want to refer to the figure on p 40 (Clockwise ACTS, or counter CAST) 8 Answer: - To find the reference angle, subtract 80 from 40 That s a 60 angle In a triangle of hpotenuse, the side opposite the 0 angle is, and the side opposite the 60 angle is That makes cos 60c =, but cosine is negative in the third quadrant, so the answer is -

34 54 CliffsStudSolver Trigonometr 9 Answer: - The angle is in quadrant IV, so subtract it from = 0 Refer back to the triangle to find the tan 0 c = = In Q-IV, tan is negative; hence the answer is - 0 Answer: or 0577 To find the reference angle, subtract 80 That makes 60 Net, if necessar, look back at the figure on p 6 ( The right triangle ) cot 60c =, which ou can rationalize: Finall, consider that the angle originall terminated in Q-III (third quadrant), where tangent, and therefore cotangent, is positive Answer: Subtract 00 from 60, and ou find that this is a 60 angle in Q-IV Secant 60 is = Since cosine is positive in Q-IV, so is secant Answer: cosi=- 5, tani=- 4, csc i= 5 4, seci=- 5, and coti =- 4 Since sini = 5 4 and cos θ is negative, the angle must be in the second quadrant Since in the first quadrant, the cosine would have been positive, and sine is positive in Q-III and Q-IV, the hpotenuse of the triangle must be 5, and the opposite side is 4 (from the sine), which makes the adjacent side, either b the Pthagorean theorem, or b remembering the Pthagorean triple, -4-5 So, cosi=- 5, tani =- 4, csc i= 5 4, seci= - 5, and coti =- 4 Answer: < 40 < 4 tan 40 = 089, tan 4 = 0869 The difference between them is = 000 $ 40 is 0, or of the wa between the two angles, so tan 40 should be 0 of the 0 difference between the two tans: = 0009 Since tan 40 is lower than tan 4 add to tan 40 : = 0848 tan 40 = Answer: < 78 < 7 sin 7 = 09455, sin 7 = 0950 The difference between them is = =

35 Chapter : Trigonometric Ideas is 08, or 8 of the wa between the two angles, so sin 78 should be 08 of the 0 difference between the two sins: = Since sin 7 is lower than sin 7, add to sin 7 : = sin 78 = Z sin5c = _ ] ) 5 Answer: 54 sin 0 i = b [ `0 04^ = the differenceh ] sin6c = b \ a Now, we ll use the variable to set up a proportion: = = = = 04 i = Supplemental Chapter Problems Solve these problems for even more practice appling the skills from this chapter The answer section will direct ou to where ou need to review Problems In which quadrant is the terminal side of a -50 angle? Name four angles that are coterminal with 70 In which quadrant does the terminal side of an angle of degree measure 400 fall? 4 What is the lowest possible positive degree measure for an angle that is coterminal with one of -950? 5 Name two positive and two negative angles that are coterminal with an angle of degree measure 5

36 56 CliffsStudSolver Trigonometr Problems 6-7 refer to the following figure β 5 α γ 6 Find cosβ 7 Find cscα 8 Find cotβ 9 An angle has a cosine of 07 What is its secant? 0 An angle has a cosecant of 8 Name its inverse function and find its value Find the sin and tangent of λ if λ is an acute angle (0 < λ < 90 ) and cos m = 5 Find the sin of θ if θ is an acute angle (0 < θ < 90 ), tanθ = 445 and cosθ = 046 The shortest leg of a triangle is 0 inches long a How long is the other leg? b How long is the hpotenuse? 4 The hpotenuse of a triangle is 0 cm long The shorter leg of the triangle is a, and the longer b Find the lengths of a and b 5 In triangle ABC, with right angle at B, the cosine of A is What are the sine of A and the tangent of A? 6 What is the sign of tan 5? 7 What is the sign of csc 0? 8 What is the cosine of 5? 9 A 5 angle is in standard position What is its secant? 0 What is the sine of an 855 angle in standard position? Angle χ is in standard position, and its terminal side passes through the point with coordinates (-8, ) Find all si of its trigonometric functions

37 Chapter : Trigonometric Ideas 57 How man seconds are there in an 8 angle? For what acute angle does cosine have a value of 0765 if cos 40 = and cos 4 = 07547? 4 Find sin 87 to the nearest ten thousandth (four decimal places) 5 For what acute angle does sine have a value of 086 if sin 55 = 089 and sin 56 = 0890? Answers I (Angles and Quadrants, p ) 40, 790, 50, 50, and so on (Coterminal Angles, p 5) IV (Coterminal Angles, p 5) 4 50 (Coterminal Angles, p 5) 5-595, -5, 485, 845, and so on (Coterminal Angles, p 5) (Trigonometric Functions of Acute Angles, p 8) 5 (Trigonometric Functions of Acute Angles, p 8) 8 5 (Reciprocal Trigonometric Functions, p 9) (Reciprocal Trigonometric Functions, p 9) 0 sine, (Reciprocal Trigonometric Functions, p 9) 5 4, 4 (Trigonometric Cofunctions, p 4) 0906 (Trigonometric Cofunctions, p 4) 0 in, 0 in (Two Special Triangles, p 5) 4 0 cm, 0 cm (Two Special Triangles, p 5) 5 5, 5 (Functions of General Angles, p 8) 6 positive (Functions of General Angles, p 8) 7 negative (Functions of General Angles, p 8) or (Reference Angles, p 4) 9 44 or (Reference Angles, p 4) or + (Reference Angles, p 4)