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Chapter 5 Trigonometric Equations 5.5 MATHPOWERTM 12, WESTERN EDITION 5.5.1 Sum and Difference Identities sin(A + B) = sin A cos B + cos A sin B sin(A - B) = sin A cos B - cos A sin B cos(A + B) = cos A cos B - sin A sin B cos(A - B) = cos A cos B + sin A sin B tan A tan B tan( A B) 1 tan Atan B tan A tan B tan( A B) 1 tan Atan B 5.5.2 Simplifying Trigonometric Expressions 1. Express cos 1000 cos 800 + sin 800 sin 1000 as a trig function of a single angle. This function has the same pattern as cos (A - B), with A = 1000 and B = 800. cos 100 cos 80 + sin 80 sin 100 = cos(1000 - 800) = cos 200 2. Express sin 3 cos 6 cos 3 sin 6 as a single trig function. This function has the same pattern as sin(A - B), with A 3 and B 6 . sin cos cos sin sin 3 6 3 6 3 6 sin 6 5.5.3 Finding Exact Values 1. Find the exact value for sin 750. Think of the angle measures that produce exact values: 300, 450, and 600. Use the sum and difference identities. Which angles, used in combination of addition or subtraction, would give a result of 750? sin 750 = sin(300 + 450) = sin 300 cos 450 + cos 300 sin 450 1 2 3 2 2 2 2 2 2 6 4 5.5.4 Finding Exact Values 2. Find the exact value for cos 150. cos 150 = cos(450 - 300) = cos 450 cos 300 + sin 450 sin300 2 3 2 1 2 2 2 2 6 2 4 5 3. Find the exact value for sin . 12 5 sin sin( ) 12 4 6 sin cos cos sin 4 6 4 6 2 3 2 1 2 2 2 2 6 2 4 3 4 12 3 4 12 2 6 12 5.5.5 Using the Sum and Difference Identities Prove cos sin . 2 cos 2 cos cos sin sin 2 2 (0)(cos ) (1)(sin ) (1)(sin ) sin . sin sin L.S. = R.S. 5.5.6 Using the Sum and Difference Identities 3 Given cos , where 0 , 5 2 find the exact value of cos( ). x 6 cos r cos( ) cos cos sin sin x=3 6 6 6 r=5 3 3 4 1 ( )( ) ( )( ) r2 = x2 + y2 5 2 5 2 3 34 10 3 34 Therefore, cos( ) . 6 10 y2 = r2 - x2 = 52 - 32 = 16 y=±4 5.5.7 Using the Sum and Difference Identities 2 4 and cos B , 3 5 where A and B are acute angles, Given sin A find the exact value of sin( A B ). sin( A B ) sin A cos B cos A sin B 2 4 5 3 ( )( ) ( )( ) 3 5 3 5 x y r A 5 2 3 B 4 3 5 8 3 5 15 Therefore, sin( A B ) 8 3 5 . 15 5.5.8 Double-Angle Identities The identities for the sine and cosine of the sum of two numbers can be used, when the two numbers A and B are equal, to develop the identities for sin 2A and cos 2A. sin 2A = sin (A + A) cos 2A = cos (A + A) = sin A cos A + cos A sin A = cos A cos A - sin A sin A = 2 sin A cos A = cos2 A - sin2A Identities for sin 2x and cos 2x: sin 2x = 2sin x cos x cos 2x = cos2x - sin2x cos 2x = 2cos2x - 1 cos 2x = 1 - 2sin2x 5.5.9 Double-Angle Identities Express each in terms of a single trig function. a) 2 sin 0.45 cos 0.45 b) cos2 5 - sin2 5 cos 2x = cos2 x - sin2 x sin 2x = 2sin x cos x cos 2(5) = cos2 5 - sin2 5 sin 2(0.45) = 2sin 0.45 cos 0.45 cos 10 = cos2 5 - sin2 5 sin 0.9 = 2sin 0.45 cos 0.45 A group of Friars opened a florist shop to help with their belfry payments. Everyone liked to buy flowers from the Men of God, Find the value of cos 2x forAxrival = 0.69. so their business flourished. florist became upset that his business was suffering because people felt compelled to buy 2 2 cos 2x = cos x - sin x from the Friars, so he2 asked the 2Friars to cut back hours or cos 2(0.69) - sin 0.69 close down.= cos The0.69 Friars refused. The florist costhem 2x = 0.1896 went to and begged that they shut down. 5.5.10 Double-Angle Identities Verify the identity tan A tan A 1 cos 2 A . sin 2A 1 (cos2 A sin2 A) 2sin Acos A 1 cos2 A sin2 A) 2 sin Acos A sin2 A sin2 A 2sin Acos A 2sin 2 A 2sin Acos A sin A cos A tan A L.S = R.S. 5.5.11 Double-Angle Identities sin 2x . Verify the identity tan x 1 cos 2x tan x 2sin x cos x 2 1 2 cos x 1 2sin x cos x 2 2cos x sin x cos x tan x L.S = R.S. 5.5.12 Double-Angle Equations 2 where 0 A 2. 2 2 For cosA = where 0 A 2, A and 7 . 2 4 4 7 7 9 15 2 A 2 n, 2 n. , , 2 A , 4 4 4 4 4 4 Find A given cosA2 = A 8 7 8 9 8 7 9 15 , 8 8 15 8 , 8 , 8 y 2 2 2 y = cos 2A 5.5.13 Double-Angle Equations 1 Find A given sin A 2 = - where 0 A 2. 2 1 7 11 For sinA = - where 0 A 2, A = and . 2 6 6 7 11 19 23 , , 2 A , 6 6 6 6 7 11 2A 2 n, 2 n. 6 6 A 7 11 19 23 , , , 12 12 12 12 y = sin 2A 7 12 11 12 19 12 23 12 1 y 2 5.5.14 Using Technology 2 tan x and predict the period. Graph the function f (x) 2 1 tan x The period is . Rewrite f(x) as a single trig function: f(x) = sin 2x 5.5.15 Identities Prove 2tan x 2 1 tan x sin 2x. 2sin xcos x 2sin x cos2x sec x 2sin x x cos 1 cos 2 x 2sin x cos 2 x cos x 1 2sin xcos x L.S. = R.S. 5.5.16 Applying Skills to Solve a Problem The horizontal distance that a soccer ball will travel, when 2 2v kicked at an angle , is given by d 0 sin cos , g where d is the horizontal distance in metres, v0 is the initial velocity in metres per second, and g is the acceleration due to gravity, which is 9.81 m/s2. a) Rewrite the expression as a sine function. Use the identity sin 2A = 2sin A cos A: 2v02 d sin cos g v02 d sin2 g 5.5.17 Applying Skills to Solve a Problem [cont’d] b) Find the distance when the initial velocity is 20 m/s. From the graph, the maximum 2 v0 distance occurs when = 0.785398. d sin2 g The maximum distance is 40.7747 m. 2 (20) The graph of sin reaches its d sin2 9.81 maximum when . 2 Distance Sin 2 will reach a maximum when 2 or . 2 4 Angle 5.5.18 Again they refused. So the florist then hired Hugh McTaggert, the biggest meanest thug in town. He went to the Friars' shop beat them up, destroyed their flowers, Suggested Questions: trashed their shop, and said that if they Pages 272-274 didn't close, he'd be back. Well, totally A the 1-16, Friars 25-35 oddclosed up shop and terrified, hid in their rooms. This proved that Hugh, B 17-24,can 37-40, 43, and only Hugh, prevent florist friars. 47, 48 5.5.19