Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
1. (a) (i) State the amplitude of 15sin2x – 5.
[1]
(ii) State the period of 15sin2x – 5.
[1]
(b)
y
4
3
2
1
–180
–135
–90
–45
0
45
90
135
180 x
–1
–2
–3
–4
The diagram shows the graph of y = |f(x)| for −180° ≤ 𝑥 ° ≤ 180° , where f(x) is a trigonometric function.
(i)
Write down two possible expressions for the trigonometric function f(x).
[2]
(ii)
State the number of solutions of the equation f(x) = 1 for −180° ≤ 𝑥 ° ≤ 180° .
[1]
0606/22/F/M/18 Q (4)
2. (a) (i) Show that
(1−sin A)(1+ sinA)
(ii) Hence solve
sin A cos A
= cot A.
(1−sin 3x)(1+ sin3x)
sin 3𝑥 cos 3𝑥
=
[2]
1
2
for 0° ≤ 𝑥 ° ≤ 180° .
(b) Solve 10 tan2 y − sec y − 1 = 0 for 0 ≤ 𝑦 ≤ 2𝜋 radians.
[4]
[5]
0606/22/F/M/18 Q (11)
1
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
3. (i) Show that cos 𝜃 cot 𝜃 + sin 𝜃 = cosec 𝜃. [3]
(ii)Hence solve cos 𝜃 cot 𝜃 + sin 𝜃 = 4 for 0° ≤ 𝑥 ° ≤ 90° . [2]
0606/22/M/J/18 Q (1)
4. (i) Show that
1
−
1
= 2 tan x sec x.[4]
1+ sin 𝑥
1
1
(ii) Hence solve the equation
−
= cosec x for 0° ≤ 𝑥 ° ≤ 360° . [4]
1− sin 𝑥
1+ sin 𝑥
0606/22/O/N/18 Q (8)
1− sin 𝑥
5. The function f is defined, for 0° ≤ 𝑥 ° ≤ 360° , by f(x) = a + b sin cx, where a, b and c are constants with
b > 0 and c > 0. The graph of y = f(x) meets the y-axis at the point (0, -1), has a period of 120° and an
amplitude of 5.
(i)
Sketch the graph of y = f(x) on the axes below.
[3]
y
6
0
0°
60°
120°
180°
240°
300°
360° x
–6
(ii)
Write down the value of each of the constants a,b and c.
a = …………………… b = ……………………. c = …………………………
[2]
0606/22/F/M/19 Q (3)
6. (a) Solve 6 sin2 x – 13 cos x = 1 for 0° ≤ 𝑥 ° ≤ 360° .
(b) (i) Show that, for −
𝜋
2
𝜋
<𝑦< ,
4 tan 𝑦
2 √1+𝑡𝑎𝑛2 𝑦
[4]
can be written in the form a sin y, where a is an integer.
[3]
(ii ) Hence solve
4 tan 𝑦
√1+𝑡𝑎𝑛2 𝑦
+ 3 = 0 for −
𝜋
2
<𝑦<
𝜋
2
radians.
[1]
0606/22/M/J/19 Q (9)
2
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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7. (i) Show that
tan 𝑥
1+sec 𝑥
+
1+sec 𝑥
(ii)Hence solve the equation
tan 𝑥
=
tan 𝑥
1+sec 𝑥
+
2
sin 𝑥
Trigonometric functions
.
[5]
1+sec 𝑥
tan 𝑥
= 1 + 3 sin x for 0 ≤ 𝑥 ≤ 180° .
[4]
0606/22/O/N/19 Q (6)
8.
f( x)
2
1
0
1
2𝜋
3
4𝜋
3
2𝜋
8𝜋
3
x
2
3
4
5
6
The diagram shows the graph of f(x) = a cos bx + c for 0 ≤ 𝑥 ≤
8𝜋
3
radians.
(a) Explain why f is a function.
[1]
(b) Write down the range of f.
[1]
(c) Find the value of each of the constants a, b and c.
[4]
0606/22/F/M/20 Q (8)
9. (a) Solve 3 cot2 x – 14 cosec x – 2 = 0 for 0° ≤ 𝑥 ≤ 360° .
sin4 𝑦 − cos4 𝑦
(b)Show that
= tan y – 2 cos y sin y.
cot 𝑦
[5]
[4]
0606/22/M/J/20 Q (8)
3
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
10. DO NOT USE A CALCULATOR IN THIS QUESTION.
In this question lengths are in centimetres.
B
2√3 +1
A
You may use the following
trigonometric ratios.
sin 30° =
30°
cos 30° =
tan 30° =
1
2
√3
2
1
√3
(a) Given that the area of the triangle ABC is 5.5 cm2, find the exact length of AC. Write your answer in
the form a + b √3, where a and b are integers.
[4]
(b) Show that BC2 = c + d √3, where c and d are integers to be found.
[4]
0606/22/O/N/20 Q (8)
11. (a) Show that
sin 𝑥 tan 𝑥
1−cos 𝑥
= 1 + sec x.
[4]
(b)Solve the equation 5 tan x – 3 cot x = 2 sec x for 0° ≤ 𝑥 ≤ 360° .
[6]
0606/22/O/N/20 Q (11)
12. The function f is defined, for 0° ≤ 𝑥 ≤ 810° , by f (x) = – 2 + cos
2𝑥
3
.
(a) Write down the amplitude of f.
[1]
(b) Find the period of f.
[2]
(c) On the axes, sketch the graph of y = f (x).
[2]
4
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
y
5
0
0°
90°
180°
270°
360°
450°
540°
630°
810° x
720°
–5
0606/22/M/J/21 Q (5)
1
+
1
= 2 cot x cosec x.
sec 𝑥+1
1
1
(b)Hence solve the equation
+
= 3 sec x for 0 < x < 360.
sec 𝑥−1
sec 𝑥+1
13. (a) Show that
sec 𝑥−1
[4]
[4]
0606/22/O/N/21 Q (3)
14. DO NOT USE A CALCULATOR IN THIS QUESTION.
All lengths in this question are in centimetres.
A
60°
6– 2
6
2
You may use the following
trigonometrical ratios.
3
sin 60°
2
1
cos 60°
2
tan60 °
3
C
B
The diagram shows triangle ABC with AC = √6 − √2, AB = √6 + √2 and angle CAB
(a) Find the exact length of BC.
√6+√2
(b) Show that sin ABC =
.
4
(c) Show the perpendicular distance from A to the line BC is 1.
60° .
[3]
[2]
[2]
0606/22/O/N/21 Q (6)
5
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
15. In this question, all angles are in radians.
(a) Solve the equation sec2𝜃 = tan 𝜃 + 3 for −𝜋 < 𝜃 < 𝜋.
𝜋
tan ∅
(b) Show that, for 0 < ∅ < ,
= sec 𝜃.
2 √1− 𝑐𝑜𝑠2 ∅
3𝜋
17
(c) Given that cosec x = − and that
< x < 2𝜋, find the exact value of cot x.
8
2
[5]
[3]
[2]
0606/22/F/M/22 Q (7)
16. (a) Show that
sin x
1−cos x
+
1−cos x
sinx
(b) Hence solve the equation
= 2 cosec x
sin x
1−cos x
+
[4]
1−cos x
sinx
= 3 sin x – 1
[4]
0606/22/O/N/22 Q (7)
17. In this question all lengths are in centimetres.
B
5 –1
x
A
5 1
C
2√5
The diagram shows triangleABC which has area 3 cm2. Angle A is acute.
a. Find the exact value of sin A.
[3]
b. Find the exact value of cos A and hence find the exact value of x.
[5]
c. Find the exact value of sin B.
[3]
0606/22/O/N Q (9)
6
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
18. 4 On the axes below, sketch the graph of y = |4 cos 2𝑥| for 0 ≤ 𝑥 ≤ 2𝜋, giving the coordinates of any
points where the graph meets the axes.
[3]
y
𝜋
0
x
0606/22/F/M/23 Q (1)
19.
0606/12/F/M/19 Q 11
20.
7
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
0606/12/O/N/19 Q 1
21.
0606/12/F/M/20 Q 10
22.
0606/12/M/J/20 Q 10
8
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
23.
0606/12/O/N/20 Q 3
9
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
24.
0606/12/F/M/21 Q 2
25.
0606/12/F/M/21 Q 8
26.
0606/12/M/J/21 Q 10
27.
0606/12/O/N/21 Q 4
10
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
28.
0606/12/O/N/21 Q 5
29.
0606/12/F/M/22 Q 3
30.
0606/12/M/J/22 Q 2
31.
0606/12/M/J/22 Q 1
11
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
32.
0606/12/O/N/22 Q 1
33.
0606/12/O/N/22 Q 4
34.
0606/12/F/M/23 Q 10
35.
0606/12/M/J/23 Q 6
36.
0606/12/O/N/23 Q 5
12
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
37.
0606/12/M/J/23 Q 1
Question 38 is written on another page.
13
Tr Naing Htet Aung (Math)
0606/ paper 1 & 2
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Trigonometric functions
38.
0606/12/O/N/23 Q 2
14