Download trigonometry3.3

Achievement Standard 90637
Trigonometry 3.3
Achievement Questions
1. Give exact values for a) cosec 90°
b) cot

  
c) cos 

 6 
4
2. Write the following as trigonometric ratios of a positive acute angle
a) sin 150°
b) cos 240°
c) tan 120°
d)
3. What are the exact values of a) cos 300°
c) tan
b) sin 315°
 12 
4. a) What is the value of tan 1   ?
 5
b) What is the value of sec ?
4
3
5

12
5. A regular pentagon is inscribed in a circle of radius 15cm. What is the length of
each side of the pentagon?
C
25
6. a) What is the value of sin 2A ?
b) What is the value of cos 2A ?
A
7
B
24
7. If sin 1 x  0.911 radians, what is the value of x ?
8. cos  90  A is equal to a) cos A
b) sin A
c) -sin A
d) sin  90  A
c) -cot A
d) tan  90  A
9. tan 180  A is equal to a) tan A
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10. Express each example as a trigonometric ratio with an acute angle in radians
 3 
 2 
 7 
a) tan 
b) sin 
c) cos 



 4 
 3 
 4 
In each of the following five problems, solve for a value of angle x in degrees (3 s.f.)
11. cos x  0.548
12. 2sin x  0.6
13. 3tan 2x  7.2
14. 4cos3x  2.5
15. 3cos  2 x  25  2
In the next five problems, solve for a value of x in radians (3 s.f.)
16. 3tan 2x  5
17. 2 sin 2x  1
18. cos 2  x  0.7   0.6


3 tan 2  x    1
2

1
20. sin 3  x  0.5  
2
19.
21. Sketch the graph of y  tan x for the interval
22. Sketch the graph of y  cos 1 2 x for the interval 0  x  4
3–
2
1
– 3
1
2
3
1
2
3
1
2
3
2
  23. Sketch the graph of y  2sin 2 x for the interval   x  
22


24. Sketch the graph of y  2 cos 2  x    1 for the interval   x  
2

25. From the sketch of the curve y  A sin B  x  C   D , find the values
of A, B, C, D
y
3
2
1
– 1


2
3
2
2
x
– 2
– 3
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1– 3
1
2
3 
2

–
22 2
26. From the sketch of the curve y  A cos B  x  C   D , find the values
of A, B, C, D
y
1
–
6– 4
1
2
3
4
5
1
2
3
3
2
– 
–
22 2

2


3
2
2
– 1
2
x
– 2
– 3
27. From the sketch of the curve y  A tan Bx  D , find the values
of A, B, C, D
y
6
5
4
3
2
1
– 
–

2

– 1

2
3
2
2
x
– 2
– 3
– 4
28. The height of sea water about the mean sea level due to tides is given by
 
h  A cos  t  , where t = time in hours after high tide and A is the height of high tide
4 
in metres.
High tide of 2.5m occurs at 2pm.
 
i.
Sketch the graph of h  A cos  t 
4 
ii.
Find out the time when the first low tide occurs after 2pm.
iii. Calculate when the height of the tide is first at 1m. Give your answer to the
nearest minute.
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29. A ferris wheel at a showground rotates clockwise
about its centre A. Passengers get on at point B. A seat
starts at point B and t seconds later it is h metres above
ground level.
 
h  8cos  t   9
 20 
i.
ii.
iii.
iv.
v.
A
B
How long does one complete rotation take?
What is the length of the radius of the wheel?
How high is B above the ground?
How high above ground level does the seat rise?
What is the height of the chair after 30 seconds?
30. A spring oscillates about a mid position. The length of the spring in mm after t
seconds is given by l  128  24sin 4t
i.
ii.
iii.
iv.
v.
How long is the spring when it is stationary?
What is the maximum length of the spring?
What is the length of the spring after 3 4 of a full cycle?
What is the period of the motion?
What is the frequency of the motion?
Merit Questions
3
5
31. If sin A  and sin B  , where A and B are acute angles, find the value of
5
13
a) sin  A  B 
b) cos  A  B  (do not use a calculator)
32. Prove each of the following 1
a) 2sin 75 sin15 
2
33. Sketch the graph of y  cot x,
    3 
b) 2 cos   sin    1
4  4 

 x  2
2
34. Sketch the graph of y  cosecx, 0  x  2
3
3
35. Given tan A  , tan B  , find the value of tan  A  B 
4
5
36. Given that cos  A  B   cos A cos B  sin A sin B , establish that cos 2 A  1  2sin 2 A
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37. What is the value of
sin 75  sin15
(without a calculator)
cos 75  cos15
38. Express as a product of two trigonometric functions a) sin5 A  sin3A
b) cos A  cos5 A
39. Express as a sum or difference of two trigonometric functions a) 2cos  x  2 y  sin 3x  4 y 
b) 2cos60 sin30
Prove the following identities
1
40. tan   cot  
sin  cos 
41.
1  tan 2 
 1  2sin 2 
1  tan 2 
42.
cos 2 A
 cos A  sin A
cos A  sin A
43.
sin 2 A
 tan A
1  cos 2 A
44. sin 2 x 
45.
46.
sin 2 x cos 2 x

 sec x
sin x
cos x
sin  3 A  2 B   sin A
cos A  cos  3 A  2 B 
47. tan x 
48.
2 tan x
1  tan 2 x
 cot  A  B 
1  cos 2 x
1  cos 2 x
cosec2  1
1

 sec2  cosec2
2
2
cos 
1  sin 
49. If u 
1  sin 
1 1  sin 
, prove that

cos 
u
cos 
Solve the following equations, for the range of angles indicated.
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50. cos 2  cos   0, in radians, 0    2
51. cos  cos

3
 sin  sin

3
 0, in radians, 0    2
52. sin 2 x  cos x  1, in radians, 0  x  2
53. cos2 2 x  3sin 2 x  3  0, in radians, -  x  
54. 2sin 2   1, 0    360
55. Find the general solutions for sin3x  sin x  0 (in radians)
56. A concrete boat ramp is at 30º to the horizontal. Jules marks the concrete at the
highest point the water reaches and his friend Verne marks the lowest level reached
by the water. The distance between the two marks is 3.2 metres.
The two budding scientists then marked the halfway point between the high and low
tide marks. They discovered that the movement of the tides was uniformly cyclic
and occurred at 12 hour intervals.
a) Assuming we start our clock when the water is at its lowest level, write a
trig equation that will model the vertical movement of the tides.
b) For what fraction of the cycle time will the water be above a mark on the
concrete exactly 1 metre below the high tide mark?
c) Where will the water level on the ramp be 10 hours after the start?
Excellence Questions
57. Find the value of x given that 2sin 1 x  
58. Given that cos 2 A  1  sin 2 A , deduce that cos 4 A  1  8sin 2 A cos 2 A
59. Write cos3A  cos  A  2 A and use the expansion of cos  A  B  to derive the
formula
cos 3 A  4 cos3 A  3cos A
 5 
60. Find the exact value of sec 

 6 
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61. Find the smallest positive value in degrees of tan 1  1.5
62. Prove the formula sin 2   cos 2   1
63. Write tan 3A as tan  A  2 A and use the expansion to derive the formula for
tan3A in terms of tan A
64. Writing tan 4 A as tan  2 A  2 A , establish the formula for tan 4A in terms of
tan A
65. From the patterns established in the previous problems write the expansion of
tan5A in terms of tan A
66. Establish the formula for tan  A  B  C  .
If A, B, C are the three angles of a triangle, show that
tan A  tan B  tan C  tan A tan B tan C
67. A triangle has adjacent sides of 5cm and 7cm with an included angle of 70°. What
is the length of the third side?
68. A triangle has sides of 5cm, 6cm and 7cm. Calculate the size of the second largest
angle. If you can’t manage that, find the size of the second smallest angle.
69. Use the sine rule to find  . Use the value of  to
find the area of the triangle.
5m
th
46
7m
70. The cuboid has dimensions 12cm  8cm  5cm
What angle does the diagonal AB make with
the horizontal?
B
8
A
5
12
71. Find the lengths of x and y.
1
3
2
x
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72. The diagram to establish the formula
for sin  A  B 
1 unit

R
S

P
U
Q
Prove the following identities
1  cos 
1  cos 
73.
 cot   cosec 
74.
1
1

 tan 2 A c osecA
cos A  sin A cos A  sin A
75.
sin 2  cos 2  1
 cot 
sin 2  cos 2  1
76.
sin 4  sin 2
 tan 2
cos 4  cos 2  1
2

77. If sin  x     cos  x    , find tan x in terms of  and  .
78. If tan 2   2 tan 2   1, show that 2cos2   cos2   0
sin 4 x  cos 4 x  1
for all values of x for which the expression
sin 4 x  cos 4 x  1
on the right hand side is defined.
For what values of x in the interval 0  x  2 is the right hand side of the
expression not defined?
79. Prove that tan 2 x 
Solve the following equations.
80. sin 3x  cos x  0 Give the general solution in radians
81. 3cot 2   5  7cosec Give the general solution in radians
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82. sin 2 x  2sin x cos x  8cos 2 x  0 Give the general solution in radians
83. sin 2 2  sin 2   1  0 , in radians, 0    
84. sin 4x  sin 2x  0 , in degrees, 180  x  180
85. cos3x  cos 2x  cos x  0 , in degrees 0  x  360
86. cos 2 2 x  3sin 2 x  3  0 , in radians 0    2
87.
2 tan 
 3 tan  . Exact values of  in the interval 0    2
1  tan 2 
88.
cos 2 x 
1
Exact values of x in the interval 0    2
2
89. A balloon is vertically over a point which lies in a direct line between two observers
a distance of 800m apart. The observers note respectively the angles of elevation of
the balloon to be 59° and 34°.
Find the height of the balloon.
90. Show that x 
Hh
H h
H
h
x
d2
91.
d1
A man stands on top of a wall of height h metres and observes the elevation of a
wireless mast to be α. He then descends from the wall and finds the elevation to be
β.
Show that the height of the mast exceeds that of the man by
h sin  cos 
metres.
sin     
h
92. h is the height of a vertical tower standing
at a point C on a horizontal plane ABC
tan  sin 
Show that h 
sin    
A


C
d

B
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93.
The angle of elevation of a tower CD from A is
15° and from B it is 19°. A and B are 60m
apart. Find the distance x.
D
h
94. Show that h 
19
15
60m
A
B
d sin  sin 
sin    
C
x
D
h

A

d
B
C
PROBLEMS
Q2 (d) missing ratio
Q21 missing interval
Do you want grids on the graphs?
Q28 ,29
I have changed to way the variable is written
Is there too much excellence? Hardly anyone does it.
Is Q56 merit or excellence?
Q60 do you need “exact value”
Q69 is angle 46 or 42.
There is a line of writing above Q66 Do I include it??
Are basic sine and cosine rule questions really excellence?????
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