Created by T. Madas
TRIGONOMETRY
EXAM QUESTIONS
INTRODUCTION
Created by T. Madas
Created by T. Madas
Question 1
(**+)
Solve the following trigonometric equation in the range given.
cos(2θ + 25)° = −0.454 , 0 ≤ θ < 360 .
C2G , θ ≈ 46, 109, 226, 289
Question 2 (**+)
Solve the following trigonometric equation in the range given.
cos(2 y − 35)° = 0.891 , 0 ≤ y < 360 .
MP1-M , y ≈ 4, 31, 184, 211
Created by T. Madas
Created by T. Madas
Question 3
(**+)
Solve the following trigonometric equation in the range given.
tan(5 y − 35)° = −2 − 3 , 0 ≤ y < 90 .
C2J , y ≈ 28, 64
Question 4 (**+)
Solve, in radians, the following trigonometric equation
1 + sin 2 x = 1 , 0 ≤ x < 2π ,
3
giving the answers correct to three significant figures.
x = 1.94c , 2.78c , 5.08c , 5.92c
Created by T. Madas
Created by T. Madas
Question 5
(**+)
Solve the following trigonometric equation in the range given.
2 cos θ = sin θ , 0° ≤ θ < 360° .
θ = 63.4°, 243.4°
Question 6 (**+)
Solve the following trigonometric equation in the range given.
2sin θ = 5cos θ , 0° ≤ θ < 360° .
C2H , x ≈ 68.2°, 248.2°
Created by T. Madas
Created by T. Madas
Question 7 (**+)
Solve the following trigonometric equation in the range given.
2sin y + 5cos y = 2cos y , 0° ≤ y < 360° .
MP1-L , y ≈ 123.7°, 303.7°
Question 8 (***)
Solve the following trigonometric equation in the range given.
3cos3 x − 1 = 0.22 , −90° ≤ x < 90° .
x ≈ −22°, 22°
Created by T. Madas
Created by T. Madas
Question 9
(***)
Solve the following trigonometric equation in the range given.
1 + 2sin(θ + 25)° = 2.532 , 0 ≤ θ < 360 .
θ ≈ 25, 105
Question 10 (***)
Solve, in radians, the following trigonometric equation
4sin 2 ψ = 15cosψ , 0 ≤ ψ < 2π ,
giving the answers correct to three significant figures.
ψ ≈ 1.32c , 4.97c
Created by T. Madas
Created by T. Madas
Question 11
(***)
Solve the following trigonometric equation in the range given.
4sin 2θ + 3cos 2θ = 0 , 0° ≤ θ < 360° .
θ = 71.6°, 161.6°, 251.6°, 341.6°
Question 12 (***)
Solve the following trigonometric equation in the range given.
2 + 2sin 3ϕ = 1 , 0° ≤ ϕ < 180° .
ϕ = 70°, 110°
Created by T. Madas
Created by T. Madas
Question 13 (***)
Solve the following trigonometric equation in the range given.
9cos 4θ + 5sin 4θ = 0 , 0° ≤ θ < 180° .
θ ≈ 29.8°, 74.8°, 119.8°, 164.8°
Question 14
(***)
Solve the following trigonometric equation in the range given.
3sin 3 y + 3 cos3 y = 0 , 0° ≤ y < 180° .
y = 50°, 110°, 170°
Created by T. Madas
Created by T. Madas
Question 15
(***)
Solve, in radians, the following trigonometric equation
6cos 2 x + sin x = 4 , 0 ≤ x < 2π ,
giving the answers correct to three significant figures.
x ≈ 0.73c , 2.41c , 3.67c , 5.76c
Question 16 (***)
Solve, in radians, the following trigonometric equation
π

5 + 2 tan  3θ +  = 3 , 0 ≤ θ < π ,
3

giving the answers in terms of π .
θ=
Created by T. Madas
5π 17π 29π
,
,
36 36
36
Created by T. Madas
Question 17 (***)
Solve, in degrees, the following trigonometric equation
3sin 2 3 x − 7 cos3 x = 5 , 0° ≤ x < 180° .
C2I , x ≈ 36.5°, 83.5°, 156.5°
Question 18 (***)
Solve, in radians, the following trigonometric equation
π

8sin  − 2 x  = 4 , 0 ≤ θ < 2π ,
3

giving the answers in terms of π .
x=
Created by T. Madas
π
3π 13π 7π
,
,
12 4 12
4
,
Created by T. Madas
Question 19 (***)
Solve the following trigonometric equation in the range given.
4sin 2 θ − cos 2 θ = 8sin θ + 3 ,
0° ≤ θ < 360° .
θ ≈ 203.6°,336.4°
Question 20 (***)
Solve, in degrees, the following trigonometric equation
sin 3 x = sin 48° , 0° ≤ x < 180° .
C2C , x = 16°, 44°,136°,164°
Created by T. Madas
Created by T. Madas
Question 21
(***)
Solve, in radians, the following trigonometric equation
cos 2 x = cos
2π
, 0 ≤ x < 2π ,
5
giving the answers in terms of π .
x=
π
5
,
4π 6π 9π
,
,
5
5
5
Question 22 (***)
Solve the following trigonometric equation in the range given.
2sin 2 x − 2 cos x − cos 2 x = 1 ,
0° ≤ x < 360° .
x ≈ 70.5°, 289.5°, x = 180°
Created by T. Madas
Created by T. Madas
Question 23
(***)
Solve the following trigonometric equation.
sin ( 3θ + 72 ) ° = cos 48° , 0 ≤ θ < 180 .
MP1-E , θ = {22, 110, 142}
Created by T. Madas
Created by T. Madas
Question 24 (***+)
Solve the following trigonometric equation in the range given.
5 + cos ( 4 y − 80 ) °
= 1.5 ,
3
0 ≤ y < 180 .
y = 50, 80, 140, 170
Question 25 (***+)
Solve the following trigonometric equation in the range given.
3 + sin 2 θ
= 3cos θ ,
cos θ − 2
0° ≤ θ < 360° .
MP1-I , θ = 120°, 240°
Created by T. Madas
Created by T. Madas
Question 26 (***+)
Solve, in radians, the following trigonometric equation
( )
sin 2 3x = 1 , 0 ≤ x < 2π ,
2
2
giving the answers in terms of π .
x=
Question 27
π π
6
,
5π 7π 3π 11π
,
,
,
2 6
6
2
6
,
(***+)
Solve the following trigonometric equation in the range given.
sin x − cos x
= 2 , 0° ≤ x < 360° .
cos x
C2M , x ≈ 71.6°, 251.6°
Created by T. Madas
Created by T. Madas
Question 28
(***+)
Solve, in radians, the following trigonometric equation
1
tan 2 ϕ
= 3 , 0 ≤ ϕ < 2π ,
giving the answers in terms of π .
C2D , ϕ =
π
5π 7π 11π
,
,
6 6
6
6
,
Question 29 (***+)
Solve, in radians, the following trigonometric equation
4sin 2 2ϕ − cos 2 2ϕ = 3 + 8sin 2ϕ , 0 ≤ ϕ < 2π ,
giving the answers correct to three significant figures.
ϕ ≈ 1.78c , 2.94c , 4.92c ,6.08c
Created by T. Madas
Created by T. Madas
Question 30 (***+)
Solve the following trigonometric equation in the range given.
3cos 2 2ϕ − 4sin 2 2ϕ = 15cos 2ϕ − 6 , 0° ≤ ϕ < 360° .
MP1-J , ϕ ≈ 40.9°, 139.1°, 220.9°, 319.1°
Created by T. Madas
Created by T. Madas
Question 31 (***+)
Solve, in radians, the following trigonometric equation
3sin 2 ψ = cos 2 ψ , 0 ≤ ψ < 2π ,
giving the answers in terms of π .
ψ=
Question 32
π 5π 7π 11π
, ,
,
6 6 6 6
(***+)
Solve, in degrees, the following trigonometric equation
tan ( 3x − 75) ° = tan 450° , 300° ≤ x < 500° .
MP1-C , x = 355°, x = 415°, x = 475°
Created by T. Madas
Created by T. Madas
Question 33
(***+)
Solve the following trigonometric equation in the range given.
5sin θ − 2cos θ
= 3,
sin θ
0° ≤ θ < 360° .
MP1-G , x = 45°, 225°
Question 34 (***+)
2CT − 2C + T − 1
a) Write the above expression as a product of two linear factors.
b) Hence solve the trigonometric equation
2 cos θ tan θ − 2cos θ + tan θ = 1 ,
for 0° ≤ θ < 360° .
MP1-D ,
( 2C + 1)(T − 1) , θ = 45°,
Created by T. Madas
120°, 135°, 240°
Created by T. Madas
Question 35 (***+)
Solve the following trigonometric equation in the range given.
cos ( 4ψ − 120 ) ° = cos 200° , 0 ≤ ψ < 180 .
MP1-H , ψ = 70, 80, 160, 170
Question 36 (***+)
Solve, in radians, the following trigonometric equation
2 + 3sin 2 4 x = 4 , 0 ≤ x <
π
2
,
giving the answers correct to three significant figures.
x = 0.239c , 0.547c , 1.02c , 1.33c
Created by T. Madas
Created by T. Madas
Question 37 (***+)
Solve the following trigonometric equation in the range given.
5cos 2 x + sin 2 x
= 7 , −90° ≤ x < 90° .
3sin 2 x
MP1-A , x ≈ −83.0°, 7.0°
Created by T. Madas
Created by T. Madas
Question 38
(***+)
Solve each of the following trigonometric equations, in the range given.
a) sin(2θ + 30°) = 3 , −180° ≤ θ < 180°
2
b) sin x = 2cos x , 0 ≤ x < 360°
c) 2sin 2 y − 5cos y + 1 = 0 , 0 ≤ y < 2π
C2B , θ = −165°, − 135°, 15°, 45° , x ≈ 63.4°, 243.4° , y =
Created by T. Madas
π
3
,
5π
3
Created by T. Madas
Question 39
(***+)
A cubic curve is given by
f ( x ) ≡ 4 x3 − 8 x 2 − x + k ,
where k is a non zero constant.
a) Given that ( x − 2 ) is a factor of f ( x ) , show that ( 2 x − 1) is also a factor of
f ( x) .
b) Express f ( x ) as the product of three linear factors.
c) Hence solve the following trigonometric equation
4sin 3 y − 8sin 2 y − sin y + k = 0 ,
for 0° ≤ y < 360° .
MP1-O , y = 30°, 150°, 210°, 330°
Created by T. Madas
Created by T. Madas
Question 40 (***+)
Solve, in radians, the following trigonometric equation
c
7 cos ( 2 x + 3) = 5 , −π ≤ x < π ,
giving the answers correct to three significant figures.
x = −1.89c , − 1.11c , 1.25c , 2.08c
Created by T. Madas
Created by T. Madas
Question 41
(***+)
The graph of the curve with equation
y = 2sin ( 2 x + k ) ° , 0 ≤ x < 360 ,
where k is a constant so that 0 < k < 90 , passes through the points with coordinates
(
)
P ( 55,1) and Q α , 3 .
a) Show, without verification, that k = 40 .
b) Determine the possible values of α .
MP1-Q , α = 10, 40,190, 220
Created by T. Madas
Created by T. Madas
Question 42 (***+)
Solve, in radians, the following trigonometric equation
c
tan ( 3 x − 5 ) = tan 7c , 3 ≤ x < 6 ,
giving the answers correct to three significant figures, where appropriate.
x = 4, x ≈ 5.05
Question 43
(***+)
Solve, in radians, the following trigonometric equation
tan 4 y − tan 2 y = 6 , 0 ≤ y < 2π ,
giving the answers in terms of π .
y=
Created by T. Madas
π 2π 4π 5π
,
,
,
3 3 3 3
Created by T. Madas
Question 44
(***+)
Solve the following trigonometric equation
2 + cos 2 x
2
3 + sin 2 x
=
2
, for 0° ≤ x < 360° .
5
MP1-U , x = 60°,120°, 240°,300°
Question 45
(***+)
Solve, in degrees, the following trigonometric equation
tan 4 y = 6 + tan 2 y , 0° ≤ y < 360° .
MP1-B , y = 60°, 120°, 240°, 300°
Created by T. Madas
Created by T. Madas
Question 46
(***+)
A trigonometric curve is defined by the equation
f ( x ) = 3 − 4sin ( 2 x + k ) ° , 0 ≤ x ≤ 360
where k is a constant such that −90 < k < 90 .
The curve passes through the point with coordinates (15,5 ) and further satisfies
A ≤ f ( x) ≤ B ,
for some constants A and B .
a) State the value of A and the value of B .
b) Show that k = −60 .
c) Solve the equation f ( x ) = −1 .
MP1-Y , A = −1 , B = 7 , x = 75, 255
Created by T. Madas
Created by T. Madas
Question 47
(****)
Given that θ is measured in degrees, solve the following trigonometric equation
4
tan 2 3θ
+2=
7
, 0 ≤ θ ≤ 180 .
sin 3θ
MP1-F , θ = 10°, 50°, 130°, 170°
Created by T. Madas
Created by T. Madas
Question 48 (****)
The depth of water in a harbour on a particular day can be modelled by the equation
 πt 
D = 12 + 3sin   ,
 6 
where D is the depth of the water in metres, t hours after midnight.
Determine the times after noon, when the depth of water in the harbour is 10 metres.
C2V , 19 : 24 , 22 : 36
Created by T. Madas
Created by T. Madas
Question 49 (****)
The height of tides in a harbour on a particular day can be modelled by the equation
h = a + b sin ( 30t ) ° ,
where h is the height of the water in metres, t hours after midnight, and a and b are
constants.
At 02.00 , h = 9.5 m and at 08.00 , h = 3.5 m .
Determine …
a) … the value of a and the exact value of b .
b) … the first time after midnight when the height of the tide is 5 metres.
MP1-W , a = 6.5 , b = 2 3 , 06 : 51
Created by T. Madas
Created by T. Madas
Question 50
(****)
Solve the following trigonometric equation, in the range given.
π
π

3 + 2sin  3 x +  = 0 , 0 ≤ x < .
2
4

Give the answers in terms of π .
C2A , x =
Question 51
13π 17π
,
36
36
(****)
Solve the following trigonometric equation in the range given.
4 tan 2 θ cos θ = 15 , 0 ≤ θ < 360° .
C2R , θ ≈ 75.5°, 284.5°
Created by T. Madas
Created by T. Madas
Question 52
(****)
Solve the following trigonometric equation in the range given.
2 tan ϕ sin ϕ = 3 , 0 ≤ ϕ < 2π .
Give the answers in terms of π .
ϕ=
Question 53
π 5π
3
,
3
(****)
Solve the following trigonometric equation in the range given.
2cos x = 3tan x , 0° ≤ x < 360° .
MP1-N , x = 30°, 150°
Created by T. Madas
Created by T. Madas
Question 54
(****)
f ( x ) = x3 − 4 x 2 − 1 x + 2 , x ∈ » .
2
a) Show that ( x − 4 ) is a factor of f ( x ) .
b) Express f ( x ) as the product of a linear and one quadratic factor.
c) Hence solve the trigonometric equation
cos3 θ − 4 cos 2 θ − 1 cos θ + 2 = 0 ,
2
for 0° ≤ θ < 360° .
(
)
C2E , f ( x ) ≡ ( x − 4 ) x 2 − 1 , θ = 45°,135°, 225°,315°
2
Created by T. Madas
Created by T. Madas
Question 55
(****)
Solve the following trigonometric equation in the range given.
2 cos x − 3tan x = 0 , 0 ≤ x < 2π .
Give the answers in terms of π .
x=
Question 56
π 5π
6
,
6
(****)
Solve the following trigonometric equation in the range given.
3tan ϕ sin ϕ = 8 , 0 ≤ ϕ < 2π .
Give the answers in radians correct to two decimal places.
ϕ ≈ 1.23c , 5.05c
Created by T. Madas
Created by T. Madas
Question 57
(****)
Solve the following trigonometric equation in the range given.
4 tanψ sinψ cosψ + 4 tanψ cosψ + 1 = 0 , 0° ≤ ψ < 360° .
C2Y , ψ = 210°, 330°
Question 58
(****)
Solve the following trigonometric equation in the range given.
1
tan x − sin x = 0 , 0° ≤ x < 360° .
2
C2Q , x = 0°, 60°, 180°, 300°
Created by T. Madas
Created by T. Madas
Question 59
(****)
Solve the following trigonometric equation in the range given.
3tan θ sin θ = cos θ + 1 , 0 ≤ θ < 2π .
Give the answers in radians correct to two decimal places.
C2F , θ ≈ 0.72c , 3.14c , 5.56c
Question 60
(****)
Solve the following trigonometric equation in the range given.
(
3 + 2sin 2 y
)(
)
3 + tan 2 y = 0 , 0 ≤ y < π .
Give the answers in terms of π .
C2L , y =
Created by T. Madas
π 2π 5π
, ,
3 3 6
Created by T. Madas
Question 61
(****)
Solve the following trigonometric equation in the range given.
6cosψ = 5 tanψ , 0 ≤ ψ < 2π .
Give the answers in radians, correct to two decimal places.
C2N , ψ ≈ 0.73c , 2.41c
Created by T. Madas
Created by T. Madas
Question 62
(****)
f ( x ) = x3 − x 2 − 3x + 3 .
a) Show that ( x − 1) is a factor of f ( x ) .
b) Express f ( x ) as the product of three linear factors.
c) Hence solve the trigonometric equation
tan 3 θ − tan 2 θ − 3tan θ + 3 = 0 ,
for 0° ≤ θ < 360° .
C2K , θ = 45°,60°,120°, 225°, 240°,300°
Created by T. Madas
Created by T. Madas
Question 63
(****)
Solve the following trigonometric equation in the range given.
3tan x + 2cos x = 0 , 0 ≤ x < 2π .
Give the answers in terms of π .
x=
Question 64
7π 11π
,
6
6
(****)
Solve the following trigonometric equation in the range given.
(
3 − 2sin 3 x
)(
)
3 + 2cos3 x = 0 , 0° ≤ x < 180° .
MP1-P , x = 20°, 50°, 70°, 140°, 160°, 170°
Created by T. Madas
Created by T. Madas
Question 65
(****)
Solve the following trigonometric equation in the range given.
8 tan 2 x sin x = cos x , 0 ≤ x < 2π .
Give the answers in radians correct to two decimal places.
C2V , x ≈ 0.46c , 3.61c
Question 66
(****+)
Solve the following trigonometric equation for 0 ≤ θ < 360°
sin θ tan 2 θ ( 2sin θ + 3) + tan 2 θ = 0 .
C2Z , θ = 0°, 180°, 210°, 330°
Created by T. Madas
Created by T. Madas
Question 67
(****+)
Calculate in degrees, correct to one decimal place, the solution of the following
trigonometric equation
1 − cos θ
= 3 sin θ , 0 < θ < π .
sin θ
MP1-X , θ ≈ 2.01c
Created by T. Madas
Created by T. Madas
Question 68
(****+)
The three angles in a triangle are denoted as α , β and γ .
It is further given that
tan α = −4.705
and
tan ( β − γ ) = 0.404
Determine, in degrees, the size of each of the angles α , β and γ .
MP1-V , α ≈ 102° , β ≈ 50° , γ ≈ 28°
Created by T. Madas
Created by T. Madas
Question 69
(****+)
y
B
y = 6 − 4sin θ − cos 2 θ
A
O
θ°
The figure above shows the graph of the curve with equation
y = 6 − 4sin θ − cos 2 θ , 0° ≤ θ ≤ 360° .
The curve has a minimum at the point A and a maximum at the point B .
Determine the coordinates of A and B .
MP1-R , A ( 90°, 2 ) , B ( 270°,10 )
Created by T. Madas
Created by T. Madas
Question 70
(*****)
Solve the following trigonometric equation for 0 ≤ θ < 360°
2 + 4 cos 2 θ = 7 cos θ sin θ .
SYN-J , θ ≈ 56.3°, θ ≈ 63.4°, θ ≈ 236.3°, θ ≈ 243.4°
Created by T. Madas
Created by T. Madas
Question 71
(*****)
It is given that
4sin x −
cos x
4
1
=
−
.
2
sin x 2cos x
Show clearly that the above equation is equivalent to
tan x = 2 .
C2T , proof
Created by T. Madas
Created by T. Madas
Question 72
(*****)
Solve the following trigonometric equation for 0 ≤ x < 360°
tan x
1
4
+
= .
cos x 1 + sin x 3
C2S , x = 30°, 150°, 210°, 330°
Created by T. Madas
Created by T. Madas
Question 73
(*****)
Solve the following trigonometric equation
(19 + 2sin 2 2θ ) tan 2θ = cos32θ − 17 cos 2θ ,
0° ≤ θ < 360° .
MP1-S , θ = 105°, θ = 165°, θ = 285°, θ = 345°
Created by T. Madas
Created by T. Madas
Question 74
(*****)
4cos 2 θ + tan 4 θ = 10 , 0 ≤ θ < 2π .
Show that θ = 1 π is a solution of the above trigonometric equation and use a non
3
verification method to find the other solutions.
MP1-T , θ = 2 π , 4 π , 5 π
3
3
3
Created by T. Madas