Trigonometric equations 6 sllabusref eferenceence Topic: Periodic functions and applications In this cha 6A 6B 6C 6D 6E chapter Simple trigonometric equations Equations using radians Further trigonometric equations Identities Using the Pthagorean identit
8 Maths Quest Maths B Year for Queensland Introduction Sudhira is a keen fisherman. The ideal depth for fishing in Sudhira s favourite tidal lake is metres. The depth of water in the lake can be found using the equation p D = sin -- t 6 where t is the time in hours after midnight. What is the best time of da for Sudhira to fish? To solve this problem we need to solve a trigonometric equation Simple trigonometric equations From our earlier work on trigonometr, ou will be familiar with problems of the tpe: Find the size of the angle marked in the figure at right. The solution to this problem is set out as: adj cos = -------- hp 9 cos = ----- = 6. cm 9 cm 9 The equation cos = ----- is an eample of a trigonometric equation. This trigonometric equation had to be solved in order to find the size of the angle in the triangle. In this particular case we knew that the angle was acute from the triangle that was drawn. In the earlier chapter on graphing periodic functions we saw that the cos function was periodic. This means that there are values of, other than the one alread found 9 for which cos = -----. There will, in fact, be an infinite number of solutions to this trigonometric equation, so for practical reasons we are usuall given a domain within which to solve the equation. This domain will often be in the form 0 60, meaning that we want solutions within the first positive revolution. If the trigonometric ratio is positive the calculator will give a first quadrant answer. To complete the solution we need to consider all quadrants for which the trigonometric ratio is positive. 9 In the case of cos = ----- the cosine ratio is positive in the first and fourth quadrants. We found earlier that the first quadrant solution to this equation was 6. The fourth quadrant solution will therefore be 60 6 =. For a negative trigonometric ratio we solve the Sine positive Tangent positive All positive Cosine positive corresponding positive equation to find a first quadrant angle to use, then find the corresponding angles in the negative quadrants.
Chapter 6 Trigonometric equations 9 Solve the following trigonometric equations over the domain 0 60, correct to the nearest degree. a sin = 0. b tan = ----- a Write the equation. a sin = 0. Use our calculator to find the first First quadrant angle = quadrant angle. 80 The sine ratio is positive in the first and second quadrants. Find the second quadrant angle b 80 = 6 subtracting from 80. Write the answer. = or 6 b Write the equation. b tan = ----- Use our calculator to find the first First quadrant angle = 0 quadrant angle. 80 0 0 60 0 The tangent ratio is negative in the second and fourth quadrants. Find the second quadrant angle b subtracting 0 from 80 and the fourth quadrant angle b subtracting 0 from 60. 80 0 = 60 60 0 = 0 Write the answer. = 60 or 0 In the earlier chapter we also found that we were able to find eact values of special angles using the triangles on the net page.
0 Maths Quest Maths B Year for Queensland 0 Line of bisection 60 These special angles should be used where possible in the solution to a trigonometric equation. The are used when we recognise an of the values produced b the triangles. sin 0 = -- cos 0 = ------ tan 0 = ------ sin = ------ cos = ------ tan = sin 60 = ------ cos 60 = -- tan 60 = Solve the equation cos q = ------ over the domain 0 60. Write the equation. cos = ------ Use the special triangles to find the first First quadrant angle = 0 quadrant angle. The cosine ratio is negative in the second and third quadrants. Find the second quadrant angle b 80 0 = 0 subtracting 0 from 80 and find the 80 + 0 = 0 third quadrant angle b adding 0 to 80. Write the answer. = 0 or 0 Similarl, we must be aware of when the boundar angles should be used in the solution of the equation. Remember from the work on the unit circle that = sin, = cos and tan = --. 80 90 0 0 or 60 70
Chapter 6 Trigonometric equations Solve the equation sin = - in the domain 0 60. Write the equation. = sin so find the angle with a -value of. sin = = 70 remember remember. Trigonometric equations are equations that use the trigonometric ratios.. The trigonometric functions are periodic and so the have an infinite number of solutions. The equation is usuall written with a restricted domain to limit the number of answers.. There are two solutions to most trigonometric equations with a domain 0 60.. Remember the special triangles as the are used in man solutions.. Boundar angles ma also provide the solution to an equation. 6A Simple trigonometric equations Solve each of the following trigonometric equations over the domain 0 60, correct to the nearest degree. a sin = 0.6 b 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 eact solutions to each of the following trigonometric equations over the domain 0 60. a sin = ------ b cos = ------ c tan = d sin = -- SkillSHEET SkillSHEET 6. 6. e cos = -- f tan = ------ g sin = ------ h cos = ------ multiple choice If sin = cos = ------ and 0 60, then is: A 0 or 0 B or C D or E 0 It is known that sin < 0 and that tan > 0. Which quadrant does the angle lie in? Eplain our answer.
Maths Quest Maths B Year for Queensland Yvonne is doing a trigonometric problem that has reduced to the equation sin =.. a When Yvonne tries to solve this equation her calculator returns an error message. Wh? b When checking her working Yvonne realises that she should have used the tangent ratio. Wh is it now possible to achieve a solution to the equation tan =.? 6 Solve each of the following equations over the domain 0 60. a sin = b cos = 0 c tan = 0 d sin = 0 e cos = f sin = 7 Solve the following trigonometric equations over the domain 60 60. a sin = 0. b cos = 0. c tan = d sin = 0.87 e cos = 0.87 f tan =. Equations using radians We have seen that a radian is an alternative method of measuring an angle. A trigonometric equation can be solved using radians as well as degrees. Usuall the domain given will indicate whether it is epected that ou will solve the equation in degrees or in radians. For eample, if ou are asked to solve an equation over the domain 0 60 then degrees are epected for the answer. However, if the given domain is 0 then it is epected that the answer will be given in radians. The method of solving the equations is the same, but be sure that our calculator is in radian mode before attempting to solve the problem to give an answer in radians. Solve the equation tan = 0.8 over the domain 0. Give the answer correct to decimal places. Write the equation. tan = 0.8 Use our calculator to find the first First quadrant angle = 0.67 quadrant angle. The tangent ratio is positive in the first and third quadrants. 0.67 + 0.67 Find the third quadrant angle b adding + 0.67 =.8 0.67 to. Write the answer. = 0.67 or.8
Chapter 6 Trigonometric equations When the special angles are used, it is still important to recognise them and recognise their radian equivalents in terms of. sin -- = -- cos -- = ------ tan -- = 6 6 6 sin -- = ------ cos -- = ------ tan -- = ------ sin -- = ------ cos -- = -- tan -- = Solve the equation sin = ------ over the domain 0. Write the equation. sin = Use the special triangles to find the first quadrant angle. The sine ratio is negative in the third and fourth quadrants. ------ First quadrant angle = -- + Find the third quadrant angle b adding -- to and the fourth quadrant angle b subtracting -- from. Write the answer. = ----- or = + -- or = + = ----- or = ----- ----- -- All of the equations that we have dealt with so far have been one-step solutions. In man eamples we ma need to rearrange the equation before we are able to use the calculator to solve it. When rearranging the equation, we attempt to place the trigonometric ratio alone on one side of the equation, as in the eample above.
Maths Quest Maths B Year for Queensland Find if sin = 0.98 over the domain 0 p. 6 Write the equation. sin = 0.98 Divide both sides b to get sin b sin = 0.9 itself. Use our calculator to find the first First quadrant angle = 0. quadrant angle. The sine ratio is positive in the first and second quadrants. 0. 0. 6 Find the second quadrant angle b = 0. subtracting 0. from. =.68 Write the answer. = 0. or.68 remember remember. Man trigonometric equations will need to be solved using radians.. The domain within which ou are asked to solve the equation will tell ou whether to use degrees or radians.. You will need to know the special angle results as the appl to radians.. You must isolate the trigonometric ratio before ou can solve an equation using either our calculator or the special angles. 6B Equations using radians Solve each of the following equations over the domain 0. Give our answers correct to decimal places. a sin = 0.8 b cos = 0. c tan =. d sin = 0.7 e cos = 0.707 f tan = 0.9 Solve each of the following over the domain 0. a sin = ------ b cos = -- c tan = d cos = e tan = f sin = -- ------
Chapter 6 Trigonometric equations Solve each of the following over the domain 0. a sin = 0 b tan = 0 c cos = 0 d sin = e cos = f cos = g sin = Find eact solutions to each of the following equations over the domain 0. 6 a sin = b cos = c tan = d sin + = 0 e cos + = 0 f tan + = 0 multiple choice The solution to the equation cos + = 0 over the domain 0 is: A --, ----- B ----- 7, ----- C --, ----- 6 6 6 6 D -----, ----- E -----, ----- 6 Solve each of the following, to the nearest degree, over the domain 0 60. a sin = b cos = c tan 7 = 0 d + sin = e + cos = f tan + 9 = 0 Further trigonometric equations In man cases the equation that we have to solve ma not be in the domain 0. We ma be asked to solve the equation in the domain 0 ( revolutions) or (also revolutions, but one in the negative sense). To find the solutions to a trigonometric equation beond the first revolution we simpl add or subtract to the first revolution solutions. Find α if sin α = 0.7 in the domain 0 α p. 7 Write the equation. sin α = 0.7 Use our calculator to find the first First quadrant angle = 0.77 quadrant angle. The sine ratio is positive in the first and second quadrants. 0.77 0.77 Continued over page
6 Maths Quest Maths B Year for Queensland 6 Find the second quadrant angle b subtracting 0.77 from. α = 0.77 α =.66 Find the solutions between and α = 0.77 + α =.66 + adding to each of the first revolution solutions. α = 7.086 α = 8.69 Write the answer. α = 0.77,.66, 7.086, 8.69 In man equations ou will first need to make the trigonometric ratio the subject of the equation. Find if cos + = 0 over the domain -. Write the equation. cos + = 0 Make cos the subject of the equation. cos = cos = ------ Use the special triangles to find the first First quadrant angle = -- quadrant angle. The cosine ratio is negative in the second and third quadrants. Find the second quadrant angle b = -- = + -- subtracting -- from. Find the third = ----- = ----- quadrant angle b adding -- to. To find the solutions between and = ----- = ----- 0, subtract from each of the first revolution solutions. = --------- = --------- Write the answer. = ---------, ---------, -----, ----- 6 7 8 remember remember. To find solutions to trigonometric equations between and we add to an solutions in the first revolution.. To find solutions to trigonometric equations between and 0 we subtract from an solutions in the first revolution.. In man cases it ma be necessar to rearrange an equation to make the trigonometric ratio the subject.
Chapter 6 Trigonometric equations 7 6C Further trigonometric equations 7 Solve each of the following trigonometric equations over the domain 0. a cos = 0.69 b sin = 0.90 c cos = 0.8 d sin = 0.7 e tan = 0.8 f tan =. Solve each of the following trigonometric equations over the domain. a sin = 0.686 b cos = 0.7 c tan = 0.776 d sin = 0. e cos = 0.97 f tan = Find the solutions to the following trigonometric equations over the domain. 8 a sin = 0 b cos = 0 c sin + = 0 d tan + = 0 e cos = f tan = 0 Find all the solutions to the following equations over the domain. Give each answer correct to decimal places. a sin + = 6 b cos = 0 c -- cos + =. d sin = e cos + = f cos + = 0 A particle moves in a straight line so that its distance, metres, from a point O is given b the equation = + sin t, where t is the time in seconds after the particle begins to move. a Find the distance from O when the particle begins to move. b Find the time when the particle first reaches O. Give our answer correct to decimal places. WorkSHEET 6. Fishing You should now be able to solve the fishing problem given at the start of this chapter. The depth of water in the lake was given b p D = sin -- t 6 Substitute D = and solve to find the best time for Sudhira to fish. The solutions should be found in the domain 0 < t <. Identities An identit is a relationship that holds true for all legitimate values of a pronumeral or pronumerals. For eample, a simple identit is + =. The identities described in this section are far more interesting and useful than this, as ou will see.
8 Maths Quest Maths B Year for Queensland The Pthagorean identit Consider the right-angled triangle in the unit circle shown. Appling Pthagoras theorem to this triangle gives the identit: sin + cos = O cos sin D P( ) The tangent Consider the unit circle on the right. A tangent is drawn at A and etended to the point C, so that OC is an etension of OP. This tangent is called tangent, which is abbreviated to tan. Triangles ODP and OAC are similar, because the have their three corresponding angles equal. O cos P( ) sin D B C A tan tan sin It follows that: ----------- = ------------ (corresponding sides) cos or tan = sin ------------ cos (as mentioned in an earlier section). Another relationship between sine and cosine complementar functions Consider the unit circle shown on the right: The triangles OAB and ODC are congruent because the have all corresponding angles equal and the hpotenuse equal (radius = ). Therefore all corresponding sides are equal and it follows that: D C(90 ) B( ) O A sin (90 ) = cos = and cos (90 ) = sin = OR sin (-- ) = cos and cos (-- ) = sin We sa that sine and cosine are complementar functions. Although the complementar function for tangent is not required for this course, ou ma like to tr to find it; that is, tan (90 ) =?
9 Chapter 6 Trigonometric equations 9 If sin = 0. and 0 < < 90, find, correct to decimal places: a cos b tan. a Use the identit sin + cos =. a sin + cos = Substitute 0. for sin. (0.) + cos = Solve the equation for cos correct to decimal places. cos = 0.6 = 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. b sin sin Use the identit tan = ------------. b tan = ------------ cos cos Substitute 0. for sin and 0.97 for 0. = ------------ cos. 0.97 Calculate the solution correct to decimal places. = 0.6 0 Find all possible values of sin if cos = 0.7. Use the identit sin + cos =. sin + cos = Substitute 0.7 for cos. sin + (0.7) = Solve the equation for sin correct to decimal places. sin = 0.6 = 0.7 sin = ± 0.7 Retain both the positive and negative solutions, since the angle could be in either the first or fourth quadrants. = 0.66 or 0.66 Find a if 0 < a < 90 and a sin a = cos b 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 b Write the equation. b cos a = sin 7 Replace sin 7 with cos (90 7). cos a = cos (90 7) cos a = cos 7 a = 7
0 Maths Quest Maths B Year for Queensland If 0 < a < 90 and cos a = --, find the eact values of: a sin a b tan a c cos (90 a) d sin (80 + a). Draw a right-angled triangle. Mark in angle a, its adjacent side (adj) and the hpotenuse (hp). hp = a adj = Use Pthagoras to calculate the opposite side (opp) to a. opp = O = = O = opp a Use the right-angled triangle to find --------. a sin a = hp Substitute opp = and hp =. = opp b Use the right-angled triangle to find --------. b tan a = adj Substitute opp = and adj =. = c Use the identit cos (90 a) = sin a. c cos (90 a) = sin a d Substitute sin a = ------. cos (90 a) = Use the smmetr propert sin (80 + a) = sin a. opp -------- hp d sin (80 + a) = sin a Substitute sin a = ------. sin (80 + a) = (Note: The above results could have been obtained using the identities directl.) ------ opp -------- adj ------ ------ ------ remember remember. sin + cos =. sin (90 ) = cos. sin -- = cos sin. tan = ------------. cos (90 ) = sin 6. sin -- = cos cos
Chapter 6 Trigonometric equations 6D Identities Cop and complete the table below, correct to decimal places: 0 8 9 9 60 0 7 sin cos sin + cos 9a 9b 0 If sin = 0.8 and 0 < < 90, find, correct to decimal places: a cos b tan. If cos = 0. and 0 < < 90, find, correct to decimal places: a sin b tan. Find all possible values of the following. a cos if sin = 0. b 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 b sin c cos. b c 6 Use the diagram at right to find the eact values of: a b b cos c tan. 8 7 7 Find the eact values of: a cos if sin = ----- and 90 < < 80 b 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: A cos B cos 6 C tan 6 D sin 6 E tan b cos is equal to: A tan 6 B cos 6 C tan D sin 6 E sin 6 c b
Maths Quest Maths B Year for Queensland c d tan 6 is equal to: A cos 6 ----------------- sin 6 B sin 6 cos 6 C D sin cos E sin 6 + cos 6 tan is equal to: A cos 6 ----------------- sin 6 B sin cos C D sin cos E sin 6 cos 6 sin 6 ----------------- cos 6 cos ----------------- sin 9 Find a if 0 a 90 and: a sin a = cos 0 b 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 table. sin 0.8 0.8 0.77 0.7 cos 0.6 0.96 0. 0.7 tan.8. 0.7 If 0 < a, b, c < 90 and sin a = ------, cos b = --, tan c = ---------, find: a sin b b tan b c cos a d tan a e sin c f cos c g sin (90 a) h cos (90 b) i sin (90 c) j sin (80 a) k cos (80 + b) 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 below are also identities using one of the following methods: i Use the identities above, and algebraic manipulation. ii Complete a table 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 both sides graphs are identical. sin = sin cos sin + sin = sin cos + tan = -------------- cos sin ( + ) = sin cos + cos sin
Chapter 6 Trigonometric equations Using the Pthagorean identit Consider the quadratic equation = 0. This equation is solved b first factorising the epression then solving each factor equal to zero. Hence, there are two solutions to the equation, as shown. = 0 ( ) = 0 so = 0 or = 0. That is, = 0 or =. A similar equation involves the use of the trigonometric ratios. Consider the equation sin = sin. This equation is solved in the same wa as a normal quadratic equation; however, the two answers are in terms of the trigonometric ratio and then have to be solved. Solve the equation sin = sin over the domain 0. 6 Write the equation. sin = sin Move sin to the left of the equation. sin sin = 0 Factorise the epression. sin ( sin ) = 0 Set each factor equal to zero and solve. sin = 0 or sin = 0 sin = -- Solve sin = 0 and sin = 0. = 0,, = --, ----- 6 6 Combine all five solutions to the = 0, --,, -----, 6 6 equation. Some equations of this tpe will involve both the sin and cos ratios, but to solve the equation there must be onl one ratio. We use the identit sin + cos =. Solve the equation sin = cos + over the domain 0. Write the equation. sin = cos + Make the substitution sin = cos. ( cos ) = cos + Form a quadratic equation b bringing all terms to one side of the equation. cos = cos + cos cos = 0 cos + cos = 0 Factorise the quadratic. ( cos )(cos + ) = 0 Continued over page
Maths Quest Maths B Year for Queensland 6 7 Solve each factor equal to 0. cos = 0 or cos = 0 cos = -- cos = Solve cos = -- and cos =. = --, ----- = Combine all solutions. = --,, ----- remember remember. Some trigonometric equations are solved as quadratic equations.. sin + cos =. Equations should have onl one trigonometric ratio. The identities cos = sin and sin = cos can be used to reduce an equation to one trigonometric ratio. 6E Using the Pthagorean identit Solve the trigonometric equation cos = cos over the domain 0. Solve each of the following equations over the domain 0. a sin sin = 0 b cos + cos = 0 c sin + sin = 0 d cos + cos = 0 e sin + sin = 0 f sin sin = 0 Solve the trigonometric equation sin = sin over the domain 0. Solve each of the following equations over the domain 0. a cos = + sin b sin + sin = 0 c sin = 0 d + cos = sin e sin = + cos f cos = + sin WorkSHEET 6.
Chapter 6 Trigonometric equations summar Simple trigonometric equations Trigonometric equations are equations that use trigonometric ratios. Trigonometric equations are periodic and so ma have an infinite number of solutions unless the domain is restricted. In a domain of one revolution most trigonometric equations will have two solutions. Be aware of the special triangles as the ma provide the solution to man equations. Radians A trigonometric equation ma need to be solved using radians. The domain within which ou are asked to solve the equation will tell ou whether to use degrees or radians. Further trigonometric equations To find solutions to a trigonometric equation between and, add to an solutions in the first revolution. To find solutions to trigonometric equations between and 0, subtract from an solutions in the first revolution. Identities sin + cos = sin tan = ------------ cos sin (90 ) = cos or sin -- = cos cos (90 ) = sin or sin -- = cos Using the Pthagorean identit The Pthagorean identit can be used to simplif a quadratic equation using two trigonometric ratios when one of them is squared. cos = sin and sin = cos
6 Maths Quest Maths B Year for Queensland CHAPTER review 6A 6A 6A 6A 6B 6B Solve the following trigonometric equations over the domain 0 60, correct to the nearest degree. a sin = 0.9 b cos = 0. c tan =.6 Find eact solutions to the following trigonometric equations over the domain 0 60. a sin = ------ b cos = -- c tan = ------ Solve each of the following equations over the domain 0 60. a sin = b cos = c tan = Solve the following trigonometric equations over the domain 60 60. a cos = 0. b tan =. c sin = 0.9 Solve each of the following equations over the domain 0. Give our answers correct to decimal places. a sin = 0.7 b cos = 0.8 c tan = 0. 6 Solve each of the following over the domain 0. a cos = ------ b tan = c sin = 6B 6B 6C 6C 6C 6D 6D 6E 6E 7 Find eact solutions to each of the following equations over the domain 0. a sin = b cos = c tan = 8 Solve each of the following, to the nearest degree, over the domain 0 60. a sin = b cos = c tan 6 = 0 9 Solve each of the trigonometric equations below over the domain 0. a cos = 0.8 b sin = 0.0 c tan = 0.8 0 Solve the trigonometric equations below over the domain. a sin = 0.86 b cos = 0.7 c tan =.677 Find solutions to the following trigonometric equations over the domain. a cos + = 0 b sin = 0 c ------ tan = 0 Find: a cos if sin = 0. and lies in the second quadrant b sin if cos = ----- and is in the third quadrant. Given that a lies in the first quadrant find a if: a sin a = cos 0 b cos a = cos 8. Solve the trigonometric equation sin = sin for the domain 0. Solve each of the following equations over the domain 0. a cos cos = 0 b sin + sin = 0 c cos = + sin d sin cos = 0 ------ test ourself CHAPTER 6