MATH131 Tutorial 1: Trigonometry
1. Calculate the area of a sector in a circle of radius 3m, with a central angle of π3 .
2.
π 5π 11π
(a) cos (b) sin (c) cot (d) csc(225)
12 3 12
23π 7π 3π 19π
(e) sec (f ) csc (g) cot (h) sin
12 4 4 12
11π 13π 7π
(i) tan − (j) cot (k) csc − (l) cot 225
6 2 4
3
3. If θ is a Quadrant II angle with sin(θ) = 5
, find cos(θ)
√
3π 5
4. If π < θ < 2
with cos(θ) = − 5
, find sin(θ)
2
5. If cos(θ) = 3
and 0 < θ < π2 , find
(a) sin 2θ (b) cos 2θ (c) tan 2θ
3 π
6. If tan α = 4
and tan β = 2, where 0 < α < 2
and 0 < β < π2 , find
(a) sin(α − β) (b) cos(α + β)
7. Find all of the angles which satisfy the given equation
1 1
(a) cos θ = (b) sin θ = − (c) cos θ = 0
2 2
8. If tan θ = 2 and 0 < θ < π2 . Find sin( 2θ )
5
9. If tan θ = − 12 and π < θ < 3π
2
. Find cos(2θ)
10. If tan A − tan B = m and cot B − cot A = n then tan(A − B) =?
cos A cos B 1
11. If = = , − π2 < A, B < 0 then 3 sin A + 6 sin B =?
3 4 5
12. If cos(α + β) = 45 , and sin(α − β) = 5
13
0 < α, β < π
4
find
(a) cot 2α (b) sin 2α (c) sin α
1
13. If sin x cos y = 8
and 2 cot x = 3 cot y then sin(x + y) =?
√ √
14. Simplify the expression sin4 x + 4 cos2 x − cos4 x + 4 sin2 x
√ √ !
π 3−1 3+1 √
15. Find all x in the interval 0 < x < 2
such that + =4 2
sin x cos x
16. If the root of the equation 2 sin2 θ + sin2 2θ = 2, 0 ≤ θ ≤ π
2
is α and β, where α < β,
then β − α =?
17. If 15 sin4 x + 10 cos4 x = 6 then tan2 x =?
15
18. If 2 sec2 x − sec4 x − 2 csc2 x + csc4 x = 4
. Find tan2 x =?
19. If the root of the quadratic equation x2 + Ax + B = 0 are tan π6 and tan π
12
, then the
value of A − B =?
20. If 15 sin4 x + 10 cos4 x = 6 then tan2 x =
21. If sin α − sin β = m and cos α − cos β = n then cos(α − β) =?
22. Find all values of x for which sin2 x = 2 sin x − 1.
√
(1 − 3) tan x
23. Find all values of x for which √ = 1.
3 − tan2 x
24. Determine the values of θ ∈ [0, 2π] for which cos 2θ = 3 cos θ − 2.
25. Determine the values of θ ∈ [0, 2π] for which cos 2θ = 2 sin θ cos θ.
26. Determine the values of θ ∈ [0, 2π] for which sin θ = cos 2θ .
27. Determine the values of θ ∈ [0, 2π] for which 3 cos θ + 3 = 2 sin2 θ.
28. Determine the values of θ ∈ [0, 2π] for which sin2 θ = 2 cos θ + 2.
√
3
29. Determine the values of θ ∈ [0, 2π] for which cos θ cos 2θ + sin θ sin 2θ = 2
.
30. (a) Express 3 sin α + 5 cos α in the form C sin α + ϕ
(b) Show that a sum of the form A sin α + B cos α can be written in the form
C sin α + ϕ .
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