Trig Problem Set 1
1. Find and prove an identity that expresses
in terms of
.
2. Give the exact value for x, where 0 ≤ x ≤ 2π .
π⎞
⎛
(a) 4 sin 2 x = 3 (b) sec ⎜ 2x + ⎟
⎝
12 ⎠
π⎞
⎛
(c) cot ⎜ x + ⎟
⎝
2⎠
(d) sin 3x =
2
2
3. The wingtip of an aeroplane is oscillating. The displacement of the wingtip is
modelled by the function
where D is the
displacement in centimetres and t is the time in seconds after the start of the
motion.
(i)
What is the largest displacement of the wingtip?
(ii)
Find the first time at which this largest displacement occurs, correct to
the nearest hundredth of a second.
(iii)
Find the first time at which a displacement of 4cm occurs, correct to
the nearest hundredth of a second.
4. Prove the following identities.
(i)
(ii)
5. Find all the values of x in the interval π ≤ x ≤ 3π for which 6 sin 2 x = 5 + cos x .
6. Prove that (a) cos ( a − b ) − cos ( a + b ) = 2 sin a sin b (b) Hence solve for x,
π⎞
π⎞
⎛
⎛
0 ≤ x ≤ 2π the equation cos ⎜ x − ⎟ = 1 + cos ⎜ x + ⎟ . (c) Find the exact value
⎝
⎝
2⎠
2⎠
of cos ( 75° ) − cos (165° ) .
7. Prove that tan15° = 2 − 3 .
8. Solve the following equations on the interval indicated.
1
(i) sin x cos x = ,−π ≤ x ≤ π
(ii)
2
9. Given the circle of radius r containing triangle
that
(the circumcircle), show
. (Draw a diagram)
10. Solve the following equation for
(i)
.
(ii)
11. On a plane hillside which slopes at an angle of 25 degrees to the horizontal, there
are two straight roads. One lies along a line of greatest slope and the other makes
an angle of 15 degrees with the horizontal. Find the angle at which the roads
intersect. (next page)
In the diagram AB and AC are segments of the two roads. Their point of
intersection A is at a height h above the point D, which lies in the horizontal plane
through BC.
12. In triangle ABC, side a = 2cm, side b = 3cm, angle A = θ and angle B = 2 θ . Find
the value of angle C and side c.
13. What is the period of the function f (x) = sin 3x + sin(6x) ?
14. The depth, h(t) metres, of water at the entrance to a harbour at t hours after
midnight on a particular day is given by h (t) = 8 + 4 sin
(a)
Find the maximum depth and the minimum depth of the water.
(b)
Find the values of t for which h (t) ³ 8.
15. The diagram below shows the boundary of the cross-section of a water
channel.
The equation that represents this boundary is y = 16 sec
– 32 where x
and y are both measured in cm.
The top of the channel is level with the ground and has a width of 24 cm. The
maximum depth of the channel is 16 cm.
Find the width of the water surface in the channel when the water depth is 10
cm. Give your answer in the form a arccos b where a, b ∈ .