7.6 Double-angle and Half-angle Formulas

Transcription

1 8 CHAPTER 7 Analytic Trigonometry. Explain why formula (7) cannot be used to show that tana p - ub cot u Establish this identity by using formulas (a) and (b). Are You Prepared? Answers.. -. (a) (b) Double-angle and Half-angle Formulas OBJECTIVES Use Double-angle Formulas to Find Exact Values (p. 8) Use Double-angle Formulas to Establish Identities (p. 8) Use Half-angle Formulas to Find Exact Values (p. 88) In this section we derive formulas for and cosa in ub, ub terms of sin uand cos u. They are derived using the sum formulas. In the sum formulas for sina + b and cosa + b, let a b u. Then sina + b cos b + cos a sin b sinu + u sin u cos u + cos u sin u sinu sin u cos u and cosa + b cos a cos b - sin b cosu + u cos u cos u - sin u sin u cosu cos u - sin u An application of the Pythagorean Identity sin u + cos u results in two other ways to express cosu. cosu cos u - sin u - sin u - sin u - sin u and cosu cos u - sin u cos u - - cos u cos u - We have established the following Double-angle Formulas: THEOREM Double-angle Formulas sinu sin u cos u cosu cos u - sin u cosu - sin u cosu cos u - () () () () Use Double-angle Formulas to Find Exact Values EXAMPLE Finding Exact Values Using the Double-angle Formulas If sin u find the exact value of:, p 6 u 6 p, (a) sinu (b) cosu

2 SECTION 7.6 Double-angle and Half-angle Formulas 8 (a) Because sinu sin u cos u and we already know that sin u we only need to find cos u. Since sin u we let y and r y r, p, 6 u 6 p, Figure y x y and place u in quadrant II. The point P x, y x, is on a circle of radius, x + y. See Figure. Then x + y (x, ) x x - y x 6 0 x We find that cos u x Now use formula () to obtain r -. sinu sin u cos u a ba- b - (b) Because we are given sin u it is easiest to use formula () to get cosu., cosu - sin u - a 9 b WARNING In finding cos(u) in Example (b), we chose to use a version of the Doubleangle Formula, formula (). Note that we are unable to use the Pythagorean Identity cos(u) ; - sin (u), with sin(u) - because we have no way of knowing, which sign to choose. Now Work PROBLEMS 7(a) AND (b) Use Double-angle Formulas to Establish Identities EXAMPLE Establishing Identities (a) Develop a formula for tanu in terms of tan u. (b) Develop a formula for sinu in terms of sin uand cos u. (a) In the sum formula for tana + b, let a b u. Then tana + b tanu + u tan a + tan b - tan a tan b tan u + tan u - tan u tan u tanu tan u - tan u () (b) To get a formula for sinu, we write uas u + uand use the sum formula. sinu sinu + u sinu cos u + cosu sin u

3 86 CHAPTER 7 Analytic Trigonometry Now use the Double-angle Formulas to get sinu sin u cos ucos u + cos u - sin usin u sin u cos u + sin u cos u - sin u sin u cos u - sin u The formula obtained in Example (b) can also be written as sinu sin u cos u - sin u sin u - sin u - sin u That is, sinu is a third-degree polynomial in the variable sin u. In fact, sinnu, n a positive odd integer, can always be written as a polynomial of degree n in the variable sin u. * Now Work PROBLEM 6 By rearranging the Double-angle Formulas () and (), we obtain other formulas that we will use later in this section. Begin with formula () and proceed to solve for sin u. cosu - sin u - cosu sin u - sin u sin u sin u - cosu (6) Similarly, using formula (), proceed to solve for cos u. cosu cos u - cos u + cosu cos u + cosu (7) Formulas (6) and (7) can be used to develop a formula for tan u. tan u sin u cos u - cosu + cosu tan u - cosu + cosu (8) Formulas (6) through (8) do not have to be memorized since their derivations are so straightforward. Formulas (6) and (7) are important in calculus. The next example illustrates a problem that arises in calculus requiring the use of formula (7). EXAMPLE Establishing an Identity Write an equivalent expression for cos uthat does not involve any powers of sine or cosine greater than. * Because of the work done by P. L. Chebyshëv, these polynomials are sometimes called Chebyshëv polynomials.

4 SECTION 7.6 Double-angle and Half-angle Formulas 87 The idea here is to apply formula (7) twice. cos u cos u a + cosu b + cosu + cos u + cosu + cos u + cosu + e + cosu f + cosu + + cosu cosu + 8 cosu Now Work PROBLEM Formula (7) Formula (7) EXAMPLE Solving a Trigonometric Equation Using Identities Solve the equation: sin u cos u - The left side of the given equation is in the form of the Double-angle Formula sin u cos u sinu, except for a factor of. Multiply each side by. sin u cos u - sin u cos u - sinu - Multiply each side by. Double-angle Formula The argument here is u. So we need to write all the solutions of this equation and then list those that are in the interval for any integer k we have, 0 u 6 p 0, p. u p + kp Because p k any integer + pkb -, u p + kp u p + - p - p, u p + 0p p, u p + p 7p, u p c c c c k - k 0 k k The solutions in the interval 0, p are + p p u p, u 7p The solution set is e p., 7p f Now Work PROBLEM 69 Figure θ EXAMPLE R Projectile Motion An object is propelled upward at an angle uto the horizontal with an initial velocity of v 0 feet per second. See Figure. If air resistance is ignored, the range R, the horizontal distance that the object travels, is given by the function Ru 6 v 0 sin u cos u (a) Show that Ru v 0 sinu. (b) Find the angle ufor which R is a maximum.

5 88 CHAPTER 7 Analytic Trigonometry (a) Rewrite the given expression for the range using the Double-angle Formula sinu sin u cos u. Then Ru 6 v 0 sin u cos u 6 v sin u cos u 0 v 0 sinu (b) In this form, the largest value for the range R can be found. For a fixed initial speed v 0, the angle uof inclination to the horizontal determines the value of R. Since the largest value of a sine function is, occurring when the argument uis 90, it follows that for maximum R we must have u 90 u An inclination to the horizontal of results in the maximum range. Use Half-angle Formulas to Find Exact Values Another important use of formulas (6) through (8) is to prove the Half-angle Formulas. In formulas (6) through (8), let u a Then. sin a - cos a cos a + cos a tan a - cos a + cos a (9) The identities in box (9) will prove useful in integral calculus. If we solve for the trigonometric functions on the left sides of equations (9), we obtain the Half-angle Formulas. THEOREM Half-angle Formulas ; - cos a A cos a ; + cos a A tan a ; - cos a A + cos a (0a) (0b) (0c) where the + or - sign is determined by the quadrant of the angle a. EXAMPLE 6 Finding Exact Values Using Half-angle Formulas Use a Half-angle Formula to find the exact value of: (a) cos (b) sin- (a) Because 0 we can use the Half-angle Formula for cos a with a 0., Also, because is in quadrant I, cos 7 0, we choose the + sign in using formula (0b): cos cos 0 + cos 0 A C + > C + +

6 SECTION 7.6 Double-angle and Half-angle Formulas 89 (b) Use the fact that sin- - sin and then apply formula (0a). sin- - sin cos 0 A - C - > - - C - - It is interesting to compare the answer found in Example 6(a) with the answer to Example of Section 7.. There we calculated cos p cos A 6 + B Based on this and the result of Example 6(a), we conclude that are equal. (Since each expression is positive, you can verify this equality by squaring each expression.) Two very different looking, yet correct, answers can be obtained, depending on the approach taken to solve a problem. Now Work PROBLEM 9 + A 6 + B and EXAMPLE 7 Finding Exact Values Using Half-angle Formulas If cos a -, p 6 a 6 p, find the exact value of: (a) (b) cos a (c) tan a First, observe that if p 6 a 6 p p a then As a result, lies in 6 a 6 p. quadrant II. a (a) Because lies in quadrant II, so use the + sign in formula (0a) 7 0, to get - cos a A R - a- b 8 R A a (b) Because lies in quadrant II, cos a so use the - sign in formula (0b) 6 0, to get cos a - + cos a A - R + a- b - R - -

7 90 CHAPTER 7 Analytic Trigonometry a (c) Because lies in quadrant II, tan a so use the - sign in formula (0c) to get 6 0, tan a - a- - - cos a A + cos a - + a- b b b b Another way to solve Example 7(c) is to use the results of parts (a) and (b). tan a sin a cos a - - Now Work PROBLEMS 7(c) AND (d) There is a formula for tan a that does not contain + and - signs, making it more useful than formula 0(c). To derive it, use the formulas - cos a sin a Formula (9) and sinca a bd cos a Double-angle Formula Then - cos a sin a cos a cos a tan a Since it also can be shown that - cos a we have the following two Half-angle Formulas: + cos a Half-angle Formulas for tan A tan a - cos a + cos a () With this formula, the solution to Example 7(c) can be obtained as follows: cos a - Then, by equation (), p 6 a 6 p - - cos a - tan a - cos a - a- - A A - b 8 - -

8 SECTION 7.6 Double-angle and Half-angle Formulas Assess Your Understanding Concepts and Vocabulary. cosu cos u sin u.. tan u. - cos u. True or False. True or False tan u tanu - tan u sinu has two equivalent forms: sin u cos u and sin u - cos u 6. True or False tanu + tanu tanu Skill Building In Problems 7 8, use the information given about the angle u, 0 u 6 p, to find the exact value of (a) sinu (b) cosu (c) sin u (d) cos u 7. sin u, 0 6 u 6 p 8. cos u, 0 6 u 6 p 9. tan u, p 6 u 6 p 0. tan u, p 6 u 6 p. cos u - 6, p 6 u 6 p. sin u -, p 6 u 6 p. sec u, sin u 7 0. csc u -, cos u 6 0. cot u -, sec u sec u, csc u tan u -, sin u cot u, cos u 6 0 In Problems 9 8, use the Half-angle Formulas to find the exact value of each expression. 9. sin. 0. cos.. tan 7p 8. cos 6. sin 9. sec p 8. tan 9p 8 6. csc 7p 8 7. sinap 8 b 8. cosa- p 8 b In Problems 9 0, use the figures to evaluate each function given that fx sin x, gx cos x, and hx tan x. y y (a, ) x y x x y x (, b) 9. fu 0. gu. ga u b. fa u b. hu. ha u b. ga 6. fa 7. fa a b 8. ga a b 9. ha a b 0. ha. Show that sin u 8 - cosu + 8 cosu.. Show that sinu cos u sin u - 8 sin u.. Develop a formula for cosu as a third-degree polynomial. Develop a formula for cosu as a fourth-degree polynomial in the variable cos u. in the variable cos u.. Find an expression for sinu as a fifth-degree polynomial 6. Find an expression for cosu as a fifth-degree polynomial in the variable sin u. in the variable cos u.

9 9 CHAPTER 7 Analytic Trigonometry In Problems 7 68, establish each identity. 7. cos u - sin u cosu cotu cot u - tan u. secu sec u - sec u cot u - tan u cot u + tan u cosu 9. cotu cot u - cot u. cos u - sin u cosu. sin u cos u - sin u sinu. 6. sin u cos u 8 - cosu 7. sec u + cos u 9. cot v sec v + sec v -. cscu sec u csc u cosu + sinu cot u - cot u + 8. csc u - cos u 60. tan v csc v - cot v 6. cos u - tan u + tan u 6. - sinu sin u + cos u sin u + cos u 6. sinu cosu sin u - cos u 6. cos u + sin u cos u - sin u - cos u - sin u cos u + sin u tanu 6. tanu tan u - tan u - tan u 66. tan u + tanu tanu + 0 tanu 67. ln ƒsin uƒ ln ƒ - cosuƒ - ln 68. ln ƒcos uƒ ln ƒ + cosuƒ - ln In Problems 69 78, solve each equation on the interval 0 u 6 p. 69. cosu + 6 sin u 70. cosu - sin u 7. cosu cos u 7. sinu cos u 7. sinu + sinu 0 7. cosu + cosu sin u cosu 76. cosu + cos u tanu + sin u tanu + cos u 0 Mixed Practice In Problems 79 90, find the exact value of each expression. 79. sina cosa 8. cosa cos - sin- sin- sinc sin- b d b b 8. tanc 8. tana 8. sina cos- tan- cos- a- 86. cosc tan - a- bd b b seca tan- cos a 90. cscc sin - a- sin- cos- b b b In Problems 9 9, find the real zeros of each trigonometric function on the interval 0 u 6 p. 9. fx sinx - sin x 9. fx cosx + cos x 9. fx cosx + sin x bd bd Applications and Extensions 9. Constructing a Rain Gutter A rain gutter is to be constructed of aluminum sheets inches wide. After marking off a length of inches from each edge, this length is bent up at an angle u. See the illustration. The area A of the opening as a function of uis given by Au 6 sin ucos u u 6 90 θ in in in in θ

10 SECTION 7.6 Double-angle and Half-angle Formulas 9 (a) In calculus, you will be asked to find the angle u that maximizes A by solving the equation (c) What is the maximum distance R if v 0 feet per second? cosu + cos u 0, 0 6 u 6 90 (d) Graph R Ru, u 90, and find the angle u that maximizes the distance R. Also find the maximum Solve this equation for u. (b) What is the maximum area A of the opening? distance. Use v 0 feet per second. Compare the results with the answers found earlier. (c) Graph A Au, 0 u 90, and find the angle u 98. Sawtooth Curve An oscilloscope often displays a sawtooth that maximizes the area A. Also find the maximum area. curve. This curve can be approximated by sinusoidal curves Compare the results to the answer found earlier. of varying periods and amplitudes. A first approximation to the sawtooth curve is given by 9. Laser Projection In a laser projection system, the optical or scanning angle uis related to the throw distance D from the scanner to the screen and the projected image width W by the equation (a) Show that the projected image width is given by u W D tan (b) Find the optical angle if the throw distance is feet and the projected image width is 6. feet. Source: Pangolin Laser Systems, Inc. 96. Product of Inertia The product of inertia for an area about inclined axes is given by the formula I uv I x sin u cos u - I y sin u cos u + I xy cos u - sin u Show that this is equivalent to I uv I x - D Source: Adapted from Hibbeler, Engineering Mechanics: Statics, 0th ed., Prentice Hall Projectile Motion An object is propelled upward at an angle u, 6 u 6 90, to the horizontal with an initial velocity of v 0 feet per second from the base of a plane that makes an angle of with the horizontal. See the illustration. If air resistance is ignored, the distance R that it travels up the inclined plane is given by the function (a) Show that Ru v 0 cos usin u - 6 θ I y W csc u - cot u sinu + R I xy cosu cos u Ru v 0 sinu - cosu - (b) In calculus, you will be asked to find the angle u that maximizes R by solving the equation sinu + cosu 0 Solve this equation for u. y sinpx + sinpx Show that y sinpx cos px. V B. Gm.V Trig TVline OH 0mv 99. Area of an Isosceles Triangle Show that the area A of an isosceles triangle whose equal sides are of length s and u is the angle between them is A s sin u [Hint: See the illustration. The height h bisects the angle and is the perpendicular bisector of the base.] 00. Geometry A rectangle is inscribed in a semicircle of radius. See the illustration. (a) Express the area A of the rectangle as a function of the angle ushown in the illustration. (b) Show that Au sinu. (c) Find the angle uthat results in the largest area A. (d) Find the dimensions of this largest rectangle. 0. If x tan u, express sinu as a function of x. 0. If x tan u, express cosu as a function of x. 0. Find the value of the number C: sin x + C - 0. Find the value of the number C: s Obase cosx cos x + C cosx h x s y u

11 9 CHAPTER 7 Analytic Trigonometry 0. If z tan a show that, 06. If z tan a show that cos a - z, + z. 07. Graph fx sin x - cosx for by using transformations. 08. Repeat Problem 07 for gx cos x. 09. Use the fact that z + z. 0 x p 0. Show that and use it to find sin p and cos p Show that cos p 8 + sin u + sin u sin u If tan u a tan u express tan u in terms of a., sinu cos p A 6 + B to find sin p and cos p. Explaining Concepts: Discussion and Writing. Go to the library and research Chebyshëv polynomials. Write a report on your findings. 7.7 Product-to-Sum and Sum-to-Product Formulas OBJECTIVES Express Products as Sums (p. 9) Express Sums as Products (p. 9) Express Products as Sums Sum and difference formulas can be used to derive formulas for writing the products of sines and/or cosines as sums or differences. These identities are usually called the Product-to-Sum Formulas. THEOREM Product-to-Sum Formulas sin b cosa - b - cosa + b cos a cos b cosa - b + cosa + b cos b sina + b + sina - b () () () These formulas do not have to be memorized. Instead, you should remember how they are derived. Then, when you want to use them, either look them up or derive them, as needed. To derive formulas () and (), write down the sum and difference formulas for the cosine: cosa - b cos a cos b + sin b cosa + b cos a cos b - sin b Subtract equation () from equation () to get () () from which cosa - b - cosa + b sin b sin b cosa - b - cosa + b