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