TRIGONOMETRY

TRIGONOMETRY
1) Use your calculator to find the value of x between 0 o and 90 o such that cos x = 1 2 .
Hence, by using a sketch graph of y = cos x, or otherwise, find ALL 4 angles in the range 360 to 360
which have a COSine of 1 2 .
2) i) By examining the symmetries of the graphs of y = sin x, y = cos x and y = tan x, determine if each of
the following properties are true for any angle . Answer true or false for each statement.
a) sin (180 ) = sin , b) cos (180 ) = cos , c) tan (180 ) = tan ,
d) sin (180 + ) = sin , e) cos (180 + ) = cos , f) tan (180 + ) = tan ,
g) sin (360 ) = sin , h) cos (360 ) = cos , i) tan (360 ) = tan ,
j) sin (90 + ) = sin (90 ), k) cos (90 + ) = cos (90 ), l) tan ( ) = tan .
ii) For each of the false properties in i), adjust the statement to give a true property; e.g. re-write b) to
give: cos (180 ) = cos .
3) Using your calculator and a sketch of the appropriate graph, find all values of x, where 0 x 360 , such
that:
a) sin x = 0 7 , b) sin x = 0 25 , c) sin x = 0
4 ,
d) cos x = 0 7 , e) cos x = 0 25 , f) cos x = 0
4 ,
g) tan x = 0 7 , h) tan x = 2 5 , i) tan x = 0
4 .
4) Find the 2 values of x between 0 o and 360 o satisfying sin x = 1 2 .
Deduce that x = 15 o is one solution to the equation sin(2x) = 1 2 and find the other solution which is less
than 180 o .
5) Sketch the graph of y = sin x for x between 0 o and 360 o . {Include the x-coordinates x = 0, 90, 180, 270
and 360.}
On the same diagram, sketch the graph of y = sin (x + 90) for x between 0 o and 360 o .
Describe the transformation which maps the graph of y = sin x onto the graph of y = sin (x + 90).
6) Sketch the graph of y = cos x for x between 0 o and 360 o . {Include the x-coordinates x = 0, 90, 180, 270
and 360.}
On the same diagram, sketch the graph of y = cos (x 30) for x between 0 o and 360 o .
Describe the transformation which maps the graph of y = cos x onto the graph of y = cos (x 30).
7) Sketch the graph of y = tan x for x between 0 o and 360 o . {Include the x-coordinates x = 0, 90, 180, 270
and 360.}
On the same diagram, sketch the graph of y = tan (x + 60) for x between 0 o and 360 o .
Describe the transformation which maps the graph of y = tan x onto the graph of y = tan (x + 60).
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8) Sketch, on the same axes, the graphs of y = cos x and y = cos 3x for x between 0 and 360. Mark clearly
the points where the graphs cross the axes.
State the period of the function y = cos 3x.
x
9) Sketch, on the same axes, the graphs of y = sin x and y = sin ( )
2
the points where the graphs cross the axes.
x
State the period of the function y = sin ( ).
2
for x between 0 and 720. Sketch clearly
10) By considering the graph of y = tan x, or otherwise, express the following in terms of tan x.
a) tan (180 + x), b) tan (180 x), c) tan (360 x).
11) The trigonometric function y = sin 2x has a period of 180.
Find the value of the constant k such that the function y = sin kx has a period of 90.
12) Find the value of the constant k such that the function y = tan kx has a period of 90.
13) Find the value of the constant k such that the function y = cos kx has a period of 45.
14) Due to tidal forces, the depth, D metres, of water in a harbour is given by D = 8 + 6sin 30t, where t is
time in hours after midnight.
i) State the maximum and minimum depths of water in the harbour at any time.
ii) How many hours elapse between instances when the water in the harbour is at its maximum
depth?
iii) What is the earliest time after midnight that the depth of water in the harbour is 12 metres? Give
your answer to the nearest minute. {Take care when converting hours to minutes!}
15) The formula P = 250 125cos 9t is used to model the number of birds in a certain remote colony
during the course of a year. P represents the number of birds, t represents time in weeks since the
beginning of the year.
i) State the period of this function.
ii) Find the greatest number of birds predicted by this model and the smallest corresponding value
of t.
iii) What is the earliest time, from the beginning of the year, when there are 300 birds in the colony?
16) Due to tidal forces, the depth, D metres, of water in a harbour is given by D = 10 + 5cos 30t, where t is
time in hours after midnight.
i) State the maximum and minimum depths of water in the harbour at any time.
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ii) How many hours elapse between instances when the water in the harbour is at its maximum
depth?
iii) What is the earliest time after midnight that the depth of water in the harbour is 12 metres? Give
your answer to the nearest minute. {Take care when converting hours to minutes!}
17) The depth, D metres, of water in a harbour is given by D = a + bcos 30t, where t is time in hours after
midnight.
Find the values of a and b such that the maximum depth is 12 metres and the minimum depth is 8
metres.
18) Find the areas of the following triangles.
12 m
a) b)
102
8 cm
7 m
52
10 cm
c)
7 m
d)
4 m
D
12 cm
30
E
8 cm
9 m
F
19) Find the area of the following triangle.
40 60
80 m
20) An equilateral triangle has area 25 cm 2 , find the length of a side.
10 cm
21) BCDEF is a regular hexagon of side 10 cm. Find its’ area.
A
B
{Hint: from the centre of the hexagon, construct 6
identical, equilateral triangles !}
F
10 cm
C
22) Find the area of a regular pentagon of side 10 cm.
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E
D

23) Without using a calculator, write down the exact values of the following trigonometric ratios.
a) sin 135 o , b) cos 240 o , c) tan 330 o , d) cos 120 o ,
e) tan 45 o , f) sin 300 o , g) sin 300 o , h) cos 150 o .
24) Solve, without using a calculator, the following equations for x between 0 o and 360 o .
a) cos x = 21
, b) tan x = 3 , c) sin x = 1
2
, d) tan x = 1
3
,
e) sin x = 21
, f) cos x = 21
, g) tan x = 1, h) sin x = 3 2
.
25) Find all 4 solutions of the equation 3tan x + 1 = 0 for 360 o x 360 o .
26) Given that sin A is negative and that cos A is positive, state, with a reason, whether the angle A is acute,
obtuse or reflex. {Hint graphs!}
27) Given that sin135
0
1
= , find, without a calculator, the value of cos 135 o .
2
28) Given that A is the obtuse angle such that sin A = 1 , find the values of cos A and tan A, expressing your
3
answers in surd form.
29) Given that A is acute and that tan A = 3 , find the values of
4
i) sin A, ii) cos A.
{Hint: consider the right angled triangle }
3
A
4
30) Given that A is not a reflex angle and that cos A = 13
5 . Find the value of tan A, giving your answer as
an exact fraction. {Hint: find sin A first.}
31) For each of the following equations, find the principal and secondary solutions and hence find all
solutions between 0 o and 360 o .
a) sin x = 0 7 , b) sin x = 0 25 , c) sin x = 0
4 ,
d) cos x = 0 7 , e) cos x = 0 25 , f) cos x = 0
4 ,
g) tan x = 0 7 , h) tan x = 2 5 , i) tan x = 0
4 .
0 0
32) Find all values of such that 180 180 , where:
a) 13sin = 5,
b) 5 tan = 2,
c) sec = 2 .
{Hint for c) rearrange to get cos = ......... etc.}
33) Solve the equation 2cos x = 3 for 180 o x 180 o .
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34) Solve the equation tan x = 2 for 0 x 360 o .
35) Solve the following equations for x between 0 o and 360 o .
1
a) 5sin 2x = 4, b) 13cos 2x = 5, c) 9tan 2x = 7, d) 10 cos x = 7,
1
1
e) 8sin x = 1,
f) 8tan x = 15,
g) 25cos 3x = 7, h) 7tan x = 24.
2
2
2
1
3
36) Solve the following equations for x between 0 o and 180 o .
1
a) 3 + 5sin 2x = 6, b) 11 + 7cos x = 8.
37) Solve the equation ( 10 )
0
1
0 0
cos + = for 180 180 .
2
+ 10 0 with y etc.}
{Hint replace ( )
2
38) Solve the following equations for x between 0 o and 360 o .
a) sin x = cos x, b) 3sin x + 4cos x = 0,
c) sin x 2 cos x = 0.
1
39) Solve the equation cos 2 = , for 0 360 0 .
4
2
40) Solve the equation 4sin 1= 0 , for 0 360 0 .
41) Solve the following equations for x between 0 o and 360 o .
2
a) 3cos
x cos x 2 = 0, b) tan 2 x tan x 6 = 0, c) tan 2 x 2tan x = 1.
42) Solve the equation sin 2 3cos 2 = 0, for between 0 o and 360 o .
{Hint: divide by cos 2 to put the equation in terms of tan 2 etc.}
2 2
43) Factorise, and hence solve the equation cos sin = 0, for 0 360 0 .
Miscellaneous questions.
44) Solve the following equations for x between 0 o and 360 o .
2
a) 3 sin x 2 cos x 2 = 0, b) 15cos 2 x = 13 + sin x, c) 10sin 2 x 5cos 2 x + 2 = 4sin x,
d) sin x = tan x, e) 5sin (2x) = cos (2x).
45) Find all values of such that 180 o < < 180 o and 2tan 3 =
2 .
tan
46) Find all values of , for 0 0 360
0 , which satisfy the equation 3sin
2 cos
2 = 2.
2
47) Use the quadratic formula to solve the equation 3z
5z
1 = 0, giving your answers to 4 decimal
places.
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2
Hence show that the solutions to 3sin
5sin
1 = 0 are given by sin = 01805 and thus
solve this equation for between 0 o and 360 o .
48) Given that A is the obtuse angle such that sin A = 5 , find the value of cos A as an exact fraction.
13
49) The diagram shows four points A, B, C, D on horizontal ground,
with AB = 500 m, AD = 600 m, CD = 250 m, angle CAB = 54 o and angle ABC = 75 o .
Calculate the length of AC, giving your answer correct to the nearest metre.
Hence show that angle ADC = 83 o correct to the nearest degree.
600 m
D
250 m
C
{Hint: SINe & COSine rules!}
A
54 o
500 m 75 o B
50) In triangle ABC, angle A = 42 o , AB = 6 cm and BC = 45
cm. Calculate the 2 possible values of angle C.
51) It is given that 0 o < < 360 o . State the two values of for which tan has the same value tan( 30 o ) .
52) The diagram shows a triangle ABC in which angle C = 30 o , BC = x cm and AC = (x + 2) cm.
Given that the area of triangle ABC is 12 cm 2 ,
calculate the value of x.
B
x cm
A
(x +2) cm
30 o C
53) Solve the equation 2sin 2 (2x) + sin (2x) 1 = 0 for x between 0 and 360. {Hint: factorise!}
54) Sketch the graph of y = cos x o , for values of x from 0 to 360.
Sketch, on the same diagram, the graph of y = cos (x 90) o .
Use your diagram to solve the equation cos x o = cos (x 90) o for values of x between 0 and 360.
Indicate clearly on your diagram how the solutions relate to the graphs.
State how many values of x satisfying the equation cos (8x) o = cos (8x 90) o lie between 0
and 360.
(You should explain your reasoning briefly, but no further detailed working or sketching is necessary.)
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55) Sketch the graph of y = sin x o , for values of x from 0 to 360.
Sketch, on the same diagram, the graph of y = sin (x + 90) o .
Use your diagram to solve the equation sin x o = sin (x + 90) o for values of x between 0 and 360.
Indicate clearly on your diagram how the solutions relate to the graphs.
State how many values of x satisfying the equation sin (12x) o = sin (12x + 90) o lie between 0
and 360.
(You should explain your reasoning briefly, but no further detailed working or sketching is necessary.)
56) Given that sin = 5
3 , and that angle is obtuse, find, without using a calculator, the exact values of
cos and tan .
57) Due to tidal forces, the depth, D metres, of water in a harbour is given by D = 4 + 3cos 30t, where t is
time in hours after midnight.
i) State the maximum and minimum depths of water in the harbour at any time.
ii) How many hours elapse between instances when the water in the harbour is at its maximum
depth?
iii) What is the earliest time after midnight that the depth of water in the harbour is 3 metres? Give
your answer to the nearest minute. {Take care when converting hours to minutes!}
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ANSWERS
1) x = 60 , 300 , 60 , 300 .
2) i) a) true, b) false, c) false, d) false, e) false, f) true, g) false, h) true, i) false, j) true, k) false, l) true.
ii) {Each false property simply requires a ‘minus’ symbol on the right-hand side!}
3) a) x = 444 , 1356 , b) x = 145 , 1655 , c) x = 3364 , 2036 , d) x = 456 , 3144 , e) x = 755 ,
2845 , f) x = 1136 , 2464 , g) x = 35 , 215 , h) x = 682 , 2482 , i) x = 3382 , 1582 .
4) x = 30 or 150 .
x = 75 .
5) A translation along the negative x-axis of 90 units.
6) A translation along the positive x-axis of 30 units.
7) A translation along the negative x-axis of 60 units.
8) 120 units.
9) 720 units.
10) a) tan x, b) tan x, c) tan x.
11) k = 4.
12) k = 2.
13) k = 8.
14) i) Maximum = 14 metres, minimum = 2 metres, ii) 12 hours, iii) 1:24 a.m.
15) i) 40 weeks, ii) 375 birds when t = 30, iii) t = 12.61979761 weeks OR approximately 12 weeks 4 days.
16) i) Maximum = 15 metres, minimum = 5 metres, ii) 12 hours, iii) 1:33 a.m.
17) a = 10, b = 2.
18) a) 315 cm 2 , b) 411 m 2 , c) 134 m 2 , d) 471 cm 2 .
19) 18083 m 2 .
20) 76 cm.
21) 2598 cm 2 .
22) 1720 cm 2 .
23) a) 1
, b)
2 21
, c) 1
3
, d) 21
, e) 1, f) 3 2
, g) 3 2
, h) 3 2
.
24) a) x = 60 o or 300 o , b) x = 60 o or 240 o , c) x = 45 o or 135 o , d) x = 30 o or 210 o , e) x = 210 o or 330 o ,
f) x = 120 o or 240 o , g) x = 135 o or 315 o , h) x = 240 o or 300 o .
25) x = 210 o or 30 o or 150 o or 330 o .
26) Reflex.
27) 1 2 .
28) cos A = 8
3
2 2
{which equals
3 }, tan A = 1 .
8
29) i) 3 5 , ii) 4 5 .
12
30) . 5
31) a) x = 44 4 0 or x = 135 6 0 , b) x = 14 5 0 or x = 165 5 0 , c) x = 203 6 0 or x = 336 4 0 ,
d) x = 45 6 0 or x = 314 4 0 , e) x = 755 0 or x = 284 5 0 , f) x = 113 6 0 or x = 246 4 0 ,
g) x = 35 0 or x = 215 0 , h) x = 68 2 0 or x = 248 2 0 , i) x = 158 2 0 or x = 338 2 0 .
32) a) =
0
22 6 or = 1574
, b) =
0
21 8 or = 1582
, c) = 60 0 or = 60
.
33) x = 30 o or 30 o .
34) x = 63.4 o or 243.4 o .
0 0 0 0
0 0 0 0
35) a) x = 266 , 634 , 2066 , 2434
, b) x = 337 , 1463 , 2137 , 3263
,
0
c) x = 189 ,
0
1089 ,
0
1989 ,
0
2889
, d) x = 268 9 0 , e) No solutions, f) x = 236 1 0 ,
g) x = 246 , 95 4 0 , 144 6 0 , 215 4 0 , 264 6 0 , 335 4
0 , h) x = 318 8 0 .
36) a) x = 184 , 716
, b) No solutions between 0 o and 180 o .
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0 0
37) x = 130 , 110 .
38) a) x = 45 o , 225 o , b) x = 1269 0 , 3069
0 , c) x = 634 0 , 2434
0 .
0 0 0 0
39) = 60 , 120 , 240 , 300 .
0 0 0 0
40) = 30 , 150 , 210 , 330 .
41) a) x = 0 0 , 360 0 , 1318
0 , 228 2
0
0 0 0 0
, b) x = 71 6 , 1166 , 2516
, 2966
,
c) x = 67.5 o , 157.5 o , 247.5 o , 337.5 o .
42) x = 30 o , 120 o , 210 o , 300 o .
0 0 0 0
43) = 45 , 225 , 135 , 315 .
44) a) x = 705 0 , 180 0 , 289 5
0 , b) x = 19.5 o , 160.5 o , 203.6 o , 336.4 o , c) x = 36.9 o , 143.1 o , 199.5 o , 340.5,
d) x = 0, 180 o , 360 o , e) x = 5.7 o , 95.7 o , 185.7 o , 275.7 o .
45) = 116.6 o , 26.6 o , 63.4 o , 153.4 o .
46) = 60 o , 120 o , 240 o , 300 o .
0 0
47) z = 18471 or 01805. = 1904 , 3496
.
48) 12
13 .
49) AC = 621 to the nearest metre.
50) C = 631 0 or 1169 0 .
51) = 150 0 or 330 0 .
52) x = 6.
53) x = 15 o , 75 o , 135 o , 195 o , 255 o , 315 o .
54) x = 45 o or x = 225 o .
55) x = 45 o or x = 225 o .
56) cos = 5
4 , tan = 4
3 .
57) i) Maximum = 7 metres, minimum = 1 metre, ii) 12 hours, iii) 3:39 a.m.
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