Download Review Problems Trig File

1.
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
 t 
h (t) = 8 + 4 sin  , 0  t  24.
6
(a)
Find the maximum depth and the minimum depth of the water.
(3)
(b)
Find the values of t for which h (t)  8.
(3)
(Total 6 marks)
2.
In a triangle ABC, Â = 35°, BC = 4 cm and AC = 6.5 cm. Find the possible values of B̂ and
the corresponding values of AB.
(Total 7 marks)
3.
A farmer owns a triangular field ABC. The side [AC] is 104 m, the side [AB] is 65 m and the
angle between these two sides is 60°.
(a)
Calculate the length of the third side of the field.
(3)
(b)
Find the area of the field in the form p 3 , where p is an integer.
(3)
Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into
two parts by constructing a straight fence [AD] of length x metres.
(c)
(i)
Show that the area o the smaller part is given by
65x
and find an expression for
4
the area of the larger part.
(ii)
Hence, find the value of x in the form q 3 , where q is an integer.
(8)
IB Questionbank Mathematics Higher Level 3rd edition
1
(d)
Prove that
BD 5
 .
DC 8
(6)
(Total 20 marks)
4.
(a)
Sketch the curve f(x) = sin 2x, 0 ≤ x ≤ π.
(2)
(b)
Hence sketch on a separate diagram the graph of g(x) = csc 2x, 0 ≤ x ≤ π, clearly stating
the coordinates of any local maximum or minimum points and the equations of any
asymptotes.
(5)
(c)
Show that tan x + cot x ≡ 2 csc 2x.
(3)
(d)
Hence or otherwise, find the coordinates of the local maximum and local minimum points
π
on the graph of y = tan 2x + cot 2x, 0 ≤ x ≤
.
2
(5)
(e)
Find the solution of the equation csc 2x = 1.5 tan x – 0.5, 0 ≤ x ≤
π
.
2
(6)
(Total 21 marks)
5.
Solve sin 2x =
2 cos x, 0 ≤ x ≤ π.
(Total 6 marks)
6.
The lengths of the sides of a triangle ABC are x – 2, x and x + 2. The largest angle is 120°.
(a)
Find the value of x.
(6)
IB Questionbank Mathematics Higher Level 3rd edition
2
(b)
Show that the area of the triangle is
15 3
.
4
(3)
(c)
Find sin A + sin B + sin C giving your answer in the form
p q
r
where p, q, r 
.
(4)
(Total 13 marks)
7.
The angle θ satisfies the equation 2 tan2 θ – 5 sec θ – 10 = 0, where θ is in the second quadrant.
Find the value of sec θ.
(Total 6 marks)
8.
Let sin x = s.
(a)
Show that the equation 4 cos 2x + 3 sin x cosec3 x + 6 = 0 can be expressed as
8s4 – 10s2 + 3 = 0.
(3)
(b)
Hence solve the equation for x, in the interval [0, π].
(6)
(Total 9 marks)
9.
(a)
If sin (x – α) = k sin (x + α) express tan x in terms of k and α.
(3)
(b)
Hence find the values of x between 0° and 360° when k =
1
and α = 210°.
2
(6)
(Total 9 marks)
IB Questionbank Mathematics Higher Level 3rd edition
3
10.
The obtuse angle B is such that tan B = 
(a)
5
. Find the values of
12
sin B;
(1)
(b)
cos B;
(1)
(c)
sin 2B;
(2)
(d0
cos 2B.
(2)
(Total 6 marks)
11.
Given that tan 2θ =
3
, find the possible values of tan θ.
4
(Total 5 marks)
IB Questionbank Mathematics Higher Level 3rd edition
4