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CHAPTER 2.5 CHAPTER 2 ANALYTICAL TRIGONOMETRY PART 5 –Multiple-Angle and Product-to-Sum Formulas TRIGONOMETRY MATHEMATICS CONTENT STANDARDS: 9.0 - Students compute, by hand, the values of the trigonometric functions and the inverse trigonometric functions at various standard points. 10.0 - Students demonstrate an understanding of the addition formulas for sines and cosines and their proofs and can use those formulas to prove and/or simplify other trigonometric identities. 11.0 - Students demonstrate an understanding of half-angle and double-angle formulas for sines and cosines and can use those formulas to prove and/or simplify other trigonometric identities. 19.0 - Students are adept at using trigonometry in a variety of applications and word problems. OBJECTIVE(S): Students will learn the double-angle formulas. Students will learn how to solve a multiple-angle equation. Students will learn how to evaluate functions involving double angles. Students will learn how to derive a multiple-angle formula. Students will learn the power-reducing formulas. Students will learn how to reduce a power. Students will learn the half-angle formulas. Students will learn how to solve a trigonometric equation using the half-angle formulas. Students will learn the product-to-sum and sum-to-product formulas. Students will learn gain further practice applying trigonometry to the real-world. Multiple-Angle Formulas In this section you will study four other categories of trigonometric identities. 1. The first category involves functions of multiple angles such as sin ku and cos ku . 2. The second category involves squares of trigonometric functions such as sin2 u . u 3. The third category involves functions of half-angles such as sin . 2 4. The fourth category involves products of trigonometric functions such as sin u cos v . F I G HJ K CHAPTER 2.5 The most commonly used multiple-angle formulas are the double-angle formulas. Double-Angle Formulas sin 2u 2 sin u cos u cos2 u sin2 u 2 cos2 u 1 1 2 sin2 u cos2u = tan 2u 2 tan u 1 tan 2 u Note that sin 2u 2 sin u , cos 2u 2 cos u , and tan 2u 2 tan u . PROOF: sin 2u = b g sin u u = = EXAMPLE 1: Solving a Multiple-Angle Equation Solve 2 cos x sin 2 x 0 . Begin by rewriting the equation so that it involves functions of x (rather than 2x). Then factor and solve as usual. 2 cos x sin 2 x 0 Write original equation. Double-angle formula. Factor. Set factors equal to zero. g Solutions in 0,2 . So, the general solution is and where n is an integer. General solution. CHAPTER 2.5 EXAMPLE 2: Using Double-Angle Formulas to Analyze Graphs Use a double-angle formula to rewrite the equation y 4 cos 2 x 2 . Using the double-angle formula for _________, you can rewrite the original equation as y 4 cos 2 x 2 Write original equation. Factor Use double-angle formula EXAMPLE 3: Evaluating Functions Involving Double Angles Use the following to find sin2 , cos2 , and tan2 . cos 5 , 13 y 3 2 2 x sin y r Consequently, using each of the double-angle formulas, you can write sin2 = = = cos2 = = = tan2 = = 5,12 CHAPTER 2.5 1.) Find sec2 . 5 12 CHAPTER 2.5 2.) If cosu 2 and u , find sin 2u . 7 2 y x The double-angle formulas are not restricted to angles 2 , and . Other double combinations, such as 4 and 2 , or 6 and 3 , are also valid. Her are two examples. sin 4 2 sin 2 cos 2 and cos 6 cos2 3 sin 2 3 CHAPTER 2.5 g 3.) Solve on the interval 0,2 : bsin 2 x cos 2 xg 1 2 4.) Simplify 6 cos2 x 3 . CHAPTER 2.5 By using double-angle formulas together with the sum formulas derived in the previous section, you can form other multiple-angle formulas. EXAMPLE 4: Deriving a Triple-Angle Formula Express sin 3x in terms of sin x . sin 3x = DAY 1 = Sum formula. = Double-angle formula. = Multiply. = Pythagorean identity. = Multiply. = Simplify. CHAPTER 2.5 Power-Reducing Formulas The double-angle formulas can be used to obtain the following power-reducing formulas. sin 2 u 1 cos 2u 2 cos2 u 1 cos 2u 2 tan 2 u 1 cos 2u 1 cos 2u EXAMPLE 5: Reducing a Power Rewrite sin4 x as a sum of first powers of the cosines of multiple angles. Note the repeated use of power-reducing formulas. sin4 x = Property of exponents. = Power-reducing formula. = Expand binomial. = Power-reducing formula. = Distributive Property. = Factor out a common factor. = Simplify. CHAPTER 2.5 Half-Angle Formulas You can derive some useful alternative forms of the power-reducing formulas by u replacing u with . The results are called half-angle formulas. 2 sin u 1 cos u 2 2 u 1 cos u cos 2 2 tan u 1 cos u sin u 2 sin u 1 cos u The signs of sin u uI F Fu I and cosGJdepend on the quadrant in which lies. G J H2 K H2 K 2 CHAPTER 2.5 EXAMPLE 6: Using a Half-Angle Formula Find the exact value of sin1050 . y x Begin by noting that 1050 is half of __________. Then, using the half-angle formula for u sin and the fact the 1050 lies in _________________________, you have 2 F I G HJ K sin1050 = = = = = The ________________ square root is chosen because sin is __________________ in Quadrant ________. CHAPTER 2.5 EXAMPLE 7: Solving a Trigonometric Equation x Find all solutions of 2 sin 2 x 2 cos2 in the interval 0,2 . 2 g 2 sin 2 x 2 cos2 x 2 Write original equation. Half-angle formula. Simplify. Simplify. Pythagorean identity. Simplify. Factor. By setting the factors _________________ and ________________________ equal to zero, you find that the solutions in the interval 0,2 are g x = ________________, x = _________________, and x = ______________. DAY 2 CHAPTER 2.5 5.) Find sin 2 . 8 15 CHAPTER 2.5 6.) Find the exact value of: 7 a. cos 12 b. tan 202.50 CHAPTER 2.5 7.) If cot u 7 and u 3 u , find cos . 2 2 y x CHAPTER 2.5 8.) Simplify 1 cos 4 x . 2 CHAPTER 2.5 g 9.) Find the exact zeros on 0,2 : x f x sin cos x 1 2 bg DAY 3 CHAPTER 2.5 Product-to-Sum Formulas Each of the following product-to-sum formulas is easily verified using the sum and difference formulas discussed in the preceding section. 1 cosu v cosu v 2 1 cos u cos v cosu v cosu v 2 1 sin u cos v sin u v sin u v 2 1 cos u sin v sin u v sin u v 2 sin u sin v EXAMPLE 8: Writing Products as Sums Rewrite the product as a sum or difference. cos 5x sin 4x cos 5x sin 4x = = Occasionally, it is useful to reverse the procedure and write a sum of trigonometric functions as a product. This can be accomplished with the following sum-to-product formulas. Sum-to-Product Formulas x y x y sin x sin y 2 sin cos 2 2 x y x y sin x sin y 2 cos sin 2 2 x y x y cos x cos y 2 cos cos 2 2 x y x y cos x cos y 2 sin sin 2 2 CHAPTER 2.5 EXAMPLE 9: Using a Sum-to-Product Formula Find the exact value of cos 1950 cos 1050 . Using the appropriate sum-to-product formula, you obtain cos 1950 cos 1050 = = = = CHAPTER 2.5 EXAMPLE 10: Solving a Trigonometric Equation Find all solutions of sin 5x sin 3x 0 in the interval 0,2 . sin 5x sin 3x 0 Write original equation. Sum-to-product formula. Simplify. Note that the general solution would be x = __________, where n is an integer. CHAPTER 2.5 EXAMPLE 11: Verifying a Trigonometric Identity sin t sin 3t tan 2t . Verify the identity cos t cos 3t Using the appropriate sum-to-product formulas, you have sin t sin 3t = cos t cos 3t = = = DAY 4 10.) Write as sum or difference: 5 sin 3 sin 4 = = = = 11.) Write as a product: cos x cos 7 x = = = CHAPTER 2.5 12.) Find the zeros on 0,2 : hx cos 2x cos 6x 13.) Verify the following trigonometric identities: u cos 4 x cos 2 x sin x a. tan csc u cot u b. 2 2 sin 3x DAY 5 CHAPTER 2.5 Application EXAMPLE 12: Projectile Motion Ignoring air resistance, the range of a projectile fired at an angle with the horizontal and with an initial velocity of v0 feet per second given by r 1 2 v0 sin cos 16 where r is the horizontal distance (in feet) that the projectile will travel. A place kicker for a football team can kick a football from ground level with an initial velocity of 80 feet per second. a.) Write the projectile motion model in a simpler form. You can use a double-angle formula to rewrite the projectile motion model as r 1 2 v0 sin cos 16 Rewrite original projectile motion model. Rewrite model using a double-angle formula. CHAPTER 2.5 b.) At what angle must the player kick the football so that the football travels 200 feet? Write projectile motion model. Substitute ______ for r and _____ for v0 . Simplify. Divide each side by 200. You know that 2 _____, so dividing this result by _____ produces _____. Because _____ = ______, you can conclude that he player must kick the football at an angle of ______ so that the football will travel 200 feet. c.) For what angle is the horizontal distance the football travels a maximum? From the model _______________________ you can see that the amplitude is ________. So the maximum range is r = _________ feet. From part b.), you know that this corresponds to angle of ______. Therefore, kicking the football at an angle of ______ will produce a maximum horizontal distance of ________ feet.