CIRCULAR FUNCTIONS/TRIGONOMETRIC EQUATIONS AND IDENTITIES. A SUMMARY OF BASIC IDENTITIES AND FORMULAE Pythagorean Identities: sin 2 cos 2 1 1 tan 2 sec 2 1 cot 2 csc2 Reciprocal and Quotient Identities: sec 1 cos csc 1 sin tan sin cos cot cos sin cot 1 tan Addition Identities: cos cos cos sin sin cos cos cos sin sin tan tan tan 1 tan tan sin sin cos cos sin sin sin cos cos sin tan tan tan 1 tan tan Double Angle Identities: cos 2 cos 2 sin 2 sin 2 2sin cos 2 cos 2 1 1 2sin 2 Formulae: x b b 2 4ac 2a Permutations and Combinations n Pr n! n r ! n Cr n Pr n n! r r ! n r ! tk 1 n Ck a nk k b 3 The angle lies in Quadrant IV with point P on the terminal arm and tan . 5 Determine the exact value of sin . 1. 2. Show all work. Answers rounded to 2 decimal places unless otherwise stated. Draw the angle 4 in standard position. *Show direction of rotation. 3 a) determine an angle that is co-terminal to b) determine the reference angle for 4 . 3 4 . 3 a) ___________________________ b) ___________________________ 3. Determine the exact value of the following. 5 a) sin 4 a) __________________________ b) cot 11 6 b)_________________________ 4. Angle lies in Quadrant II with point A 4, 6 on the terminal arm. Point P is the point of intersection of the terminal arm of and the unit circle centered at 0, 0 . Determine the exact value of the x -coordinate of point P . 4. 5. Determine the period of each function. a) y tan 1 x 2 b) y cos 2 x 3 __________________________ 6. 7. Determine the domain and range of the function: y 3sin 5 x 2 4 . Domain: __________________________ Range: __________________________ Determine the amplitude, period, phase shift (indicate direction for phase shift) and vertical displacement for the following function: y 7 cos 3x 5 4 amplitude: __________________________ period: __________________________ phase shift: __________________________ vertical displacement: 8. __________________________ Graph at least 1 period for each function on the grid provided. y 3sin x 2 2 4 2 2 4 10. Solve the following equations algebraically. Give solutions as exact values where 0 x 2 . a) 2 tan x cos x tan x a) _________________________ b) 2sin 2 x sin x 1 0 b) __________________________ 3 marks 11. Prove the following identity. cot x sec x cos x sin 2 x 2 cos x