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Trigonometry
Question Paper 6
Level
International A Level
Subject
Maths
Exam Board
CIE
Topic
Trigonometry
Sub Topic
Booklet
Question Paper 6
Time Allowed:
82 minutes
Score:
/68
Percentage:
/100
Grade Boundaries:
A*
>85%
A
777.5%
B
C
D
E
U
70%
62.5%
57.5%
45%
<45%
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1
(i) Express 5 cos θ − sin θ in the form R cos(θ + α ), where R > 0 and 0◦ < α < 90◦ , giving the exact
[3]
value of R and the value of α correct to 2 decimal places.
(ii) Hence solve the equation
5 cos θ − sin θ = 4,
giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ .
2
[4]
(i) Express 8 sin θ − 15 cos θ in the form R sin(θ − α ), where R > 0 and 0◦ < α < 90◦ , giving the
[3]
exact value of R and the value of α correct to 2 decimal places.
(ii) Hence solve the equation
8 sin θ − 15 cos θ = 14,
giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ .
3
[4]
(i) Prove the identity
tan(x + 45◦ ) − tan(45◦ − x) ≡ 2 tan 2x.
[4]
(ii) Hence solve the equation
tan(x + 45◦ ) − tan(45◦ − x) = 2,
for 0◦ ≤ x ≤ 180◦ .
4
(i) Prove the identity
[3]
√
cos(x + 30◦ ) + sin(x + 60◦ ) ≡ ( 3) cos x.
[3]
(ii) Hence solve the equation
cos(x + 30◦ ) + sin(x + 60◦ ) = 1,
for 0◦ < x < 90◦ .
[2]
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5
(i) Express 12 cos θ − 5 sin θ in the form R cos(θ + α ), where R > 0 and 0◦ < α < 90◦ , giving the
[3]
exact value of R and the value of α correct to 2 decimal places.
(ii) Hence solve the equation
12 cos θ − 5 sin θ = 10,
giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ .
6
[4]
Find the values of x satisfying the equation
3 sin 2x = cos x,
for 0◦ ≤ x ≤ 90◦ .
7
[4]
(i) Express 3 sin θ + 4 cos θ in the form R sin(θ + α ), where R > 0 and 0◦ < α < 90◦ , giving the value
[3]
of α correct to 2 decimal places.
(ii) Hence solve the equation
3 sin θ + 4 cos θ = 4.5,
giving all solutions in the interval 0◦ ≤ θ ≤ 360◦ , correct to 1 decimal place.
(iii) Write down the least value of 3 sin θ + 4 cos θ + 7 as θ varies.
8
[4]
[1]
√
(i) Express cos θ + ( 3) sin θ in the form R cos(θ − α ), where R > 0 and 0 < α < 12 π , giving the
exact value of α .
[3]
(ii) Hence show that one solution of the equation
√
√
cos θ + ( 3) sin θ = 2
is θ =
7
π,
12
and find the other solution in the interval 0 < θ < 2π .
[4]
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9
(i) Show that the equation
tan(45◦ + x) = 4 tan(45◦ − x)
can be written in the form
3 tan2 x − 10 tan x + 3 = 0.
[4]
(ii) Hence solve the equation
tan(45◦ + x) = 4 tan(45◦ − x),
for 0◦ < x < 90◦ .
10
[3]
The angle x, measured in degrees, satisfies the equation
cos(x 30Æ ) 3 sin(x 60Æ ).
(i) By expanding each side, show that the equation may be simplified to
(2
3) cos x sin x.
[3]
(ii) Find the two possible values of x lying between 0Æ and 360Æ .
[3]
(iii) Find the exact value of cos 2x, giving your answer as a fraction.
[3]