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Math 12 Pre-calculus Double/Half angles of trig functions 4 3 and cos K then find the exact values of sin(2 K ) and cos(2 K ) 5 5 1 2. Given that sin( m2 ) . 2 1. If sin K Where are the possible locations of angle m? Is it possible to definitively determine the value of sin( m) ? a) cos( m2 ) 0 . Find the exact value of tan(m) 7 5. If is acute and cos(2 ) , find the exact values of cos( ) and sin( ) 9 b) It is now known that 6. Use an appropriate formula to re-write as a single trig function a) 4cos(x)sin(x) b) 2sin2(x) – 1 c) 1 – 2cos2(3x) d) cos ( 2x ) sin ( 2x ) 2 2 7. Prove each identity: a) b) cos x sin x cos 2 x (sin x cos x)2 1 sin(2 x) 4 9. Solve on the interval 0 x 2 a) sin(2x) + sin(x) = 0 4 b) cos(2x) + 3cos(x) = 1 10. Evaluate exactly 16 8 a) sin 11. If sin b) sin 3 and is a 4th quadrant angle, find the exact value of (be careful with signs) 5 b) sin a) cos 2 2 12. Consider the diagram below to be accurate; e.g. angle A appears to be between and 2 A 4 B (450 and 900). Assuming that each angle is less than one full rotation, find the sign (+/-) of a) c) e) g) i) sin(2A) tan(2C) sin(A+B) tan(A-C) sin( D 2 b) cos(2B) d) sin(2D) f) cos(D-C) h) cos( C2 ) D C j) tan( C2 ) ) Answers for most…. 24 25 257 1/9 root(3) 0, 23 , , 43 1 4 15 4 10 10 3 , , 53 31010 4 95 18 3 87 2 2 2 (why two answers?) 1 3 2 2 3 2sin(2 x) 2 2 2 (you needed to find 2 cos(2 x) cos(6 x) cos( x) cos 8 , which is 2 2 2 ) Math 12 Pre-calculus Trigonometric Identities Primary Identities 1 csc x sin x 1 sec x cos x 1 cot x tan x Quotient Identities tan x sin x cos x cot x cos x sin x Pythagorean Identities sin 2 x cos 2 x 1 1 cot 2 x csc 2 x tan 2 x 1 sec 2 x Even / Odd Identities: Sin( ) and Tan( ) are odd, Cos( ) is even sin( x) sin x cos( x) cos x tan( x) tan x Compound Angle Identities: sin( A B) sin A cos B sin B cos A sin( A B) sin A cos B sin B cos A cos( A B) cos A cos B sin A sin B cos( A B) cos A cos B sin A sin B tan( A B) tan A tan B 1 tan A tan B tan( A B) tan A tan B 1 tan A tan B Double and Half Angle Identities: cos(2 x) cos 2 x sin 2 x or sin(2 x) 2sin x cos x cos(2 x) 2 cos 2 x 1 tan(2 x) or 2 tan x 1 tan 2 x cos(2 x) 1 2sin 2 x With the sine and cosine half-angle formulae, you will need to draw a picture of the original angle (x) and where the half-angle (x/2) is, and use it to decide if you want the + or – root. sin 2x 1 cos x 2 cos 2x 1 cos x 2 tan 2x 1 cos x sin x