6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities called double-angle and half-angle identities. We can derive these identities directly from the sum and difference identities given in Section 6-. Even though the names use the word angle, the new identities hold for real numbers as well. Double-Angle Identities Start with the sum identity for sine, sin ( y) sin cos y cos sin y and replace y with to obtain sin ( ) sin cos cos sin On simplification, this gives sin sin cos Double-angle identity for sine () If we start with the sum identity for cosine, cos ( y) cos cos y sin sin y and replace y with, we obtain cos ( ) cos cos sin sin On simplification, this gives cos cos sin First double-angle identity for cosine () Now, using the Pythagorean identity sin cos (3) in the form cos sin (4) and substituting it into equation (), we get cos sin sin On simplification, this gives cos sin Second double-angle identity for cosine (5)
47 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Or, if we use equation (3) in the form sin cos and substitute it into equation (), we get cos cos ( cos ) On simplification, this gives cos cos Third double-angle identity for cosine (6) Double-angle identities can be established for the tangent function in the same way by starting with the sum formula for tangent (a good eercise for you). We list the double-angle identities below for convenient reference. DOUBLE-ANGLE IDENTITIES sin sin cos cos cos sin sin cos tan tan cot tan cot cot tan The identities in the second row can be solved for sin and cos to obtain the identities sin cos cos cos These are useful in calculus to transform a power form to a nonpower form. Eplore/Discuss (A) Discuss how you would show that, in general, sin sin cos cos tan tan (B) Graph y sin and y sin in the same viewing window. Conclusion? Repeat the process for the other two statements in part A. Identity Verification Verify the identity cos tan. tan
6-3 Double-Angle and Half-Angle Identities 473 Verification We start with the right side: tan tan sin cos sin cos cos sin cos sin cos sin cos Quotient identities Algebra Pythagorean identity Double-angle identity Key Algebraic Steps in Eample a b a b b a b a b b b a b a Solution Verify the identity sin Finding Eact Values tan tan. Find the eact values, without using a calculator, of sin and cos if tan 3 4 and is a quadrant IV angle. First draw the reference triangle for and find any unknown sides: r 4 3 r (3) 4 5 sin 3 5 cos 4 5 Now use double-angle identities for sine and cosine: sin sin cos ( 3 5 )( 4 5 ) 4 5 cos cos ( 4 5 ) 7 5 Find the eact values, without using a calculator, of cos and tan if sin and is a quadrant II angle. 4 5
474 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS Half-Angle Identities Half-angle identities are simply double-angle identities stated in an alternate form. Let s start with the double-angle identity for cosine in the form cos m sin m Now replace m with / and solve for sin (/) [if m is twice m, then m is half of m think about this]: cos sin sin cos sin cos Half-angle identity for sine (7) where the choice of the sign is determined by the quadrant in which / lies. To obtain a half-angle identity for cosine, start with the double-angle identity for cosine in the form cos m cos m and let m / to obtain cos cos Half-angle identity for cosine (8) where the sign is determined by the quadrant in which / lies. To obtain a half-angle identity for tangent, use the quotient identity and the half-angle formulas for sine and cosine: Thus, tan sin cos cos cos cos cos tan cos cos Half-angle identity for tangent (9) where the sign is determined by the quadrant in which / lies. Simpler versions of equation (9) can be obtained as follows: tan cos cos cos cos cos cos (0)
cos ( cos ) sin ( cos ) sin ( cos ) sin cos 6-3 Double-Angle and Half-Angle Identities 475 sin sin and ( cos ) cos, since cos is never negative. All absolute value signs can be dropped, since it can be shown that tan (/) and sin always have the same sign (a good eercise for you). Thus, tan sin cos Half-angle identity for tangent () By multiplying the numerator and the denominator in the radicand in equation (0) by cos and reasoning as before, we also can obtain tan cos sin Half-angle identity for tangent () We now list all the half-angle identities for convenient reference. HALF-ANGLE IDENTITIES sin cos cos cos tan cos sin cos cos cos sin where the sign is determined by the quadrant in which / lies. Eplore/Discuss (A) Discuss how you would show that, in general, sin cos tan sin cos tan (B) Graph y sin and y sin in the same viewing window. Conclusion? Repeat the process for the other two statements in part A.
476 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS 3 Finding Eact Values Compute the eact value of sin 65 without a calculator using a half-angle identity. Solution sin 65 sin 330 cos 330 (3/) 3 Use half-angle identity for sine with a positive radical, since sin 65 is positive. 3 4 Solution Compute the eact value of tan 05 without a calculator using a half-angle identity. Finding Eact Values Find the eact values of cos (/) and cot (/) without using a calculator if sin 3 5, 3/. Draw a reference triangle in the third quadrant, and find cos. Then use appropriate half-angle identities. a a 5 (3) 4 cos 4 5 3 5 (a, 3) If 3/, then 3 4 Divide each member of 3/ by. Thus, / is an angle in the second quadrant where cosine and cotangent are negative, and cos cos (4 5 ) 0 or 0 0 cot tan (/) sin cos 3 5 ( 4 5 ) 3
6-3 Double-Angle and Half-Angle Identities 477 4 5 Verification 5 Find the eact values of sin (/) and tan (/) without using a calculator if cot 4 3, /. Identity Verification Verify the identity: sin tan sin tan sin Half-angle identity for sine cos sin cos tan cos tan tan tan cos tan tan sin tan Verify the identity cos Square both sides. Algebra Algebra tan sin. tan Quotient identity Answers to Matched Problems tan tan sin cos cos sin cos sin cos sin cos sin cos sin. sin cos cos sin cos. cos 7 3. 4. 5. cos 5, tan 4 cos tan 7 3 sin (/) 30/0, tan (/) 3 cos tan tan cos tan sin tan tan tan EXERCISE 6-3 A 5. sin cos, (Choose the correct sign.) In Problems 6, verify each identity for the values indicated.. cos cos sin, 30. sin sin cos, 45 3. tan cot tan, 3 4. tan tan tan, 6 6. cos cos, (Choose the correct sign.) In Problems 7 0, find the eact value without a calculator using double-angle and half-angle identities. 7. sin.5 8. tan 75 9. cos 67.5 0. tan 5
478 6 TRIGONOMETRIC IDENTITIES AND CONDITIONAL EQUATIONS In Problems 4, graph y and y in the same viewing window for. Use TRACE to compare the two graphs.. y cos, y cos sin. y sin, y sin cos 3. 4. B y tan, y y tan, y Verify the identities in Problems 5 3. 5. (sin cos ) sin 6. sin (tan )( cos ) 7. sin ( cos ) 8. cos (cos ) 9. cos tan sin 0. sin t (sin t cos t). sin cos. cos 3. cot tan 4. cot tan 5. cot 6. cot sin cos sin cos tan tan 7. cos u tan u 8. tan u 9. 30. sec sec csc tan tan sec cot tan 3. cos tan (/) 3. cos tan (/) cot tan Compute the eact values of sin, cos, and tan using the information given in Problems 33 36 and appropriate identities. Do not use a calculator. 33. sin 3 5, / 34. cos 5, / 35. 36. tan 5, / 0 cot 5, / 0 In Problems 37 40, compute the eact values of sin (/), cos (/), and tan (/) using the information given and appropriate identities. Do not use a calculator. 37. sin 3, 3/ 38. cos 4, 3/ 39. cot 3 4, / 40. tan 3 4, / cos cot tan cos sin cos u tan u sin u tan u 4 Suppose you are tutoring a student who is having difficulties in finding the eact values of sin and cos from the information given in Problems 4 and 4. Assuming you have worked through each problem and have identified the key steps in the solution process, proceed with your tutoring by guiding the student through the solution process using the following questions. Record the epected correct responses from the student. (A) The angle is in what quadrant and how do you know? (B) How can you find sin and cos? Find each. (C) What identities relate sin and cos with either sin or cos? (D) How would you use the identities in part C to find sin and cos eactly, including the correct sign? (E) What are the eact values for sin and cos? 4. Find the eact values of sin and cos, given tan, 0 90. 4. Find the eact values of sin and cos, given sec, 0 90. Verify each of the following identities for the value of indicated in Problems 43 46. Compute values to five significant digits using a calculator. (A) (B) cos cos tan tan tan (Choose the correct sign.) 43. 5.06 44. 7.358 45. 0.934 57 46. 4 In Problems 47 50, graph y and y in the same viewing window for, and state the intervals for which the equation y y is an identity. 47. 48. 49. 50. C 4 3 5 4 y cos (/), y cos y cos (/), y cos y sin (/), y y sin (/), y Verify the identities in Problems 5 54. 5. cos 3 4 cos 3 3 cos 5. sin 3 3 sin 4 sin 3 cos cos 53. cos 4 8 cos 4 8 cos 54. sin 4 (cos )(4 sin 8 sin 3 )
6-3 Double-Angle and Half-Angle Identities 479 In Problems 55 60, find the eact value of each without using a calculator. 3 5 )] 55. cos [ cos ( 3 5 )] 56. sin [ cos ( 57. tan [ 3 cos ( 4 5 )] 58. tan [ tan (4 )] 59. cos [ 60. sin [ tan ( 4 cos ( 3 5 )] 3 )] (B) Using the resulting equation in part A, determine the angle that will produce the maimum distance d for a given initial speed v 0. This result is an important consideration for shot-putters, javelin throwers, and discus throwers. In Problems 6 66, graph f() in a graphing utility, find a simpler function g() that has the same graph as f(), and verify the identity f() g(). [Assume g() k A T(B) where T() is one of the si trigonometric functions.] 6. f() csc cot 6. f() csc cot cos cos 63. f() 64. f() sin cos 65. f() 66. f() cot cot sin cos APPLICATIONS 70. Geometry. In part (a) of the figure, M and N are the midpoints of the sides of a square. Find the eact value of cos. [Hint: The solution uses the Pythagorean theorem, the definition of sine and cosine, a half-angle identity, and some auiliary lines as drawn in part (b) of the figure.] M M 67. Indirect Measurement. Find the eact value of in the figure; then find and to three decimal places. [Hint: Use cos cos.] s N s / / N 68. Indirect Measurement. Find the eact value of in the figure; then find and to three decimal places. [Hint: Use tan ( tan )/( tan ).] 69. Sports Physics. The theoretical distance d that a shotputter, discus thrower, or javelin thrower can achieve on a given throw is found in physics to be given approimately by d 4 feet v 0 sin cos 3 feet per second per second 8 m where v 0 is the initial speed of the object thrown (in feet per second) and is the angle above the horizontal at which the object leaves the hand (see the figure). (A) Write the formula in terms of sin by using a suitable identity. feet 7 m s (a) s (b) 7. Area. An n-sided regular polygon is inscribed in a circle of radius R. (A) Show that the area of the n-sided polygon is given by A n nr sin n [Hint: (Area of a triangle) ( )(base)(altitude). Also, a double-angle identity is useful.] (B) For a circle of radius, complete Table, to five decimal places, using the formula in part A: T A B L E n 0 00,000 0,000 A n (C) What number does A n seem to approach as n increases without bound? (What is the area of a circle of radius?) (D) Will A n eactly equal the area of the circumscribed circle for some sufficiently large n? How close can A n be made to get to the area of the circumscribed circle? [In calculus, the area of the circumscribed circle is called the limit of A n as n increases without bound. In symbols, for a circle of radius, we would write lim A n n. The limit concept is the cornerstone on which calculus is constructed.]