Name _____________________________
Topic 3 Problem Set: Trigonometry
Part I – No Calculator (Questions 1-9)
1. Given that sin x =
(a)
cos x;
(b)
cos 2x.
1
, where x is an acute angle, find the exact value of
3
(Total 6 Marks)
2. Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c.
3. If 𝑠𝑖𝑛𝑥 = −
1
√3
and 𝜋 < 𝑥 <
3𝜋
2
(2 Marks)
, find 𝑡𝑎𝑛𝑥.
(4 Marks)
4
4. If A is an obtuse angle in a triangle and sin A = , calculate the exact value of sin 2A.
5
(4 Marks)
5. Given the values of sin  and cos  , determine the quadrant in which  lies.
7
3
sin    , cos  
4
4
(2 Marks)
8. Evaluate and express your answer in simplified form.
𝑠𝑖𝑛
√3
5𝜋
6
+ 𝑐𝑜𝑠𝜋 + 𝑡𝑎𝑛
1
5𝜋
(4 Marks)
4
9. If ( 2 , − 2) is a coordinate on the unit circle find the value of the angle 𝜃.
(2 Marks)
Part II – Calculator Permitted
10. The diagram below shows a circle of radius r and centre O. The angle AÔB = θ.
The length of the arc AB is 24 cm. The area of the sector OAB is 180 cm2.
Find the value of r and of θ.
(6 Marks)
11. The following diagram shows a circle of centre O, and radius 15 cm. The arc ACB subtends
an angle of 2 radians at the centre O.
C
A
B
15
cm
Diagram not to scale
2 rad
O
AÔB = 2 radians
OA = 15 cm
Find
(a)
the length of the arc ACB;
(b)
the area of the shaded region.
(6 Marks)
12. Find the acute angle between the line given and the x-axis.
𝑦 = 4𝑥 + 1
13. Consider the equation 3 cos 2x + sin x = 1
(a)
Write this equation in the form f (x) = 0 , where f (x) = p sin2 x + q sin x + r ,
and p , q , r є .
(b)
Factorize f (x).
(c)
Write down the number of solutions of f (x) = 0, for 0 ≤ x ≤ 2π.
14. The following diagram shows a triangle ABC, where BC = 5 cm, B̂ = 60°, Ĉ = 40°.
A
B
40°
60°
(a)
Calculate AB.
(b)
Find the area of the triangle.
5 cm
C
15. [Show your work for this problem on a separate sheet of paper and attach.]
The following diagram shows two semi-circles. The larger one has centre O and radius
4 cm. The smaller one has centre P, radius 3 cm, and passes through O. The line (OP)
meets the larger semi-circle at S. The semi-circles intersect at Q.
(a)
(i)
Explain why OPQ is an isosceles triangle.
(ii)
Use the cosine rule to show that cos OP̂Q =
(iii) Hence show that sin OP̂Q =
1
.
9
80
.
9
(iv) Find the area of the triangle OPQ.
(7)
(b)
Consider the smaller semi-circle, with centre P.
(i)
Write down the size of OP̂Q.
(ii)
Calculate the area of the sector OPQ.
(3)
(c)
Consider the larger semi-circle, with centre O. Calculate the area of the sector
QOS.
(3)
(d)
Hence calculate the area of the shaded region.
(4)
(Total 17 marks)
16. [Show your work for this problem on a separate sheet of paper and attach.]
A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m, a
second side, [AB], is 65 m and the angle between these two sides is 60°.
(a)
Use the cosine rule to calculate the length of the third side of the field.
(3)
(b)
Given that sin 60° =
3
, find the area of the field in the form p 3 where p is
2
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 A1 and A2 by constructing a straight fence [AD] of length x
metres, as shown on the diagram below.
C
104 m
A2
A
30°
D
x
30°
A1
65 m
B
(c)
65x
.
4
(i)
Show that the area of Al is given by
(ii)
Find a similar expression for the area of A2.
(iii) Hence, find the value of x in the form q 3 , where q is an integer.
(7)
(Total 13 marks)
17.
The following graph shows the depth of water, y metres, at a point P, during one day.
The time t is given in hours, from midnight to noon.
(a)
Use the graph to write down an estimate of the value of t when
(i)
the depth of water is minimum;
(ii)
the depth of water is maximum;
(iii)
the depth of the water is increasing most rapidly.
(3)
(b)
The depth of water can be modelled by the function y = A cos (B (t – 1)) + C.
(i)
Show that A = 8.
(ii)
Write down the value of C.
(iii)
Find the value of B.
(6)
(c)
A sailor knows that he cannot sail past P when the depth of the water is less than 12 m.
Calculate the values of t between which he cannot sail past P.
(2)
(Total 11 marks)
 x
18. The diagram below shows the graph of f (x) = 1 + tan   for −360  x  360.
2
(a)
On the same diagram, draw the asymptotes.
(2)
(b)
Write down
(i)
the period of the function;
(ii)
the value of f (90).
(2)
(c)
Solve f (x) = 0 for −360  x  360.
(2)
(Total 6 marks)
19.
Consider g (x) = 3 sin 2x.
(a)
Write down the period of g.
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
(1)
(b)
On the diagram below, sketch the curve of g, for 0  x  2.
y
4
3
2
1
0
–1
–2
–3
–4
π
2
π
3π
2
2π
x
(3)
(c)
Write down the number of solutions to the equation g (x) = 2, for 0  x  2.
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..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
(2)
(Total 6 marks)