The University of Zambia Department of Mathematics and Statistics MAT 1120 - Introductory Mathematics, Statistics and Probability Tutorial Sheet 5 - Trignometry May, 2022 1. Write each of the following as a trigonometry ratio of positive acute angles. (a) sin(−194◦ ) (b) cos 481◦ (c) tan 185◦ (d) sin 331◦ 2. Find the exact value of the following leaving your answer in surd form where necessary. (a) sin 150◦ (b) sin(−60◦ ) (c) cos 225◦ (d) cos 240◦ (e) tan 210◦ (f) cos 7π 6 (g) sin 9π 4 (h) tan 14π 3 (i) cos − 5π 3 3. Given that θ = π3 , find the value of each of the following. (a) sin (2θ) (b) 2 sin θ (c) cos θ 2 (d) tan2 θ 4. (a) Given that tan x = − 43 and x is obtuse, find the exact value of sec x and cos x. √ (b) Given that p sin x = 4 and p cos x = 4 3 where p > 0, find the value of p and the smallest positive value of x. √ (c) Given that sin A = 23 and cos B = exact value of each of the following: √ 3 , 2 where A is obtuse and B is acute, find the (ii) cos (A − B) (i) sin (A + B) (iii) cot (A + B) 5. Find the exact value of each of the following: (a) sin 15◦ (b) cos 165◦ (c) tan 105◦ π (d) cos 12 (e) sin 11π 12 (f) tan 5π 12 6. Prove the following identities. (a) tan2 x + 1 = sec2 x (b) sec x − cos x = sin x tan x (c) (sin θ + cos θ)2 = 1 + sin 2θ x 1 − cos x (e) = tan2 1 + cos x 2 (d) (2 cos θ) (2 cos θ − 1) = 2 cos 2θ + 1 (g) (f) 1 1 − = 2 tan x sec x 1 − sin x 1 + sin(x) 1 sin 2x + sin x = tan x 1 + cos 2x + cos x (h) (sin x + cos x)(1 − sin x cos x) = sin3 x + cos3 x Tutorial Sheet 5 - Trignometry- Page 2 of 2 May, 2022 7. Solve for 0 ≤ x < 2π each of the following equations. (a) 2 cos x − 1 = 0 (b) 2 sin x cos x + cos x = 0 (c) 2 sin 3x − 1 = 0 (d) 3 csc2 x = 4 (e) 2 sin2 − cos x = 1 (f) sin 2x = 2 tan 2x (g) 10 cos2 x + cos x = 11 sin2 x − 9 (h) 3 cos2 x + 1 = 4 sin2 x 8. Find all solutions of the equation in the interval [0, 360◦ ). (c) 2 sin x tan x − tan x = 1 − 2 sin x (b) 2 sin2 x − sin x − 1 = 0 √ (d) 3 sin 2x = cos 2x (e) 7 sin(2x + 30◦ ) = 3 cos(2x + 30◦ ) (f) 5 cos 2x + 1 = sin 2x tan 2x (a) 2 cos(3x) = 1 9. Sketch the graph of each of the following functions. (b) y = 2 cos x − 21 π (a) f (x) = 1 + sin x2 (c) y = 1 − |cos (2x)| 10. State the period, the amplitude and the phase shift of each function below and sketch the curve. (a) y = −2 cos x2 (b) y = 1 + 31 sin 2 x + π2 (c) f (x) = 2 sin π x − 41 − 1 11. Given that the range of the functions sin−1 (x), cos−1 (x) and tan−1 (x) are − π2 , π2 , [0, π] and − π2 , π2 , respectively, evaluate the following, giving the answer in radians. √ √ (a) sin−1 23 (b) tan−1 − 3 (c) cos−1 √ − 3 2 (d) cos−1 √ − 2 2 12. Given that the range of the functions sin−1 (x) and cos−1 (x) are − π2 , π2 and [0, π], respectively, evaluate the following. (a) sin cos−1 12 (b) cos sin−1 21 13. Prove each of the following. x √25 − x2 (a) sin cos−1 = 5 5 x x (b) tan sin−1 =√ 2 4 − x2 14. (a) Using the trigonometric identity for tan(A + B), prove that tan 3x = 3 tan x − tan3 x , x 6= (2n + 1)30o , n ∈ Z. 1 − 3 tan2 x (b) Hence solve, for −30o < x < 30o , the equation tan 3x = 11 tan x. End of Tutorial Sheet