Unit Circle Chapter 4 Review
1. Express the following in radians, in terms of  .
(a) 340 =
(b) - 135
2. Express the following in radians, correct to the nearest tenth.
(b) 230
(a) 218 =
3. Convert the following to degrees, rounded to the nearest whole degree.
(a)
2
3
(b)
7
4
(c) 
4
5
4. Convert the following to degrees, correct to the nearest whole degree.
(a) 1.6 rad
(b) 3 
(c) 1
5. Use the formula a  r , to find the arc length, radius or angle measure in the following
problems. Remember that  must be in radians for this formula to work!!
(a) An angle at the centre of a circle that has a radius of 2 cm is subtended by an arc
long. Find the measure of the angle, correct to the nearest tenth of a radian.
3 cm
(b) On a circle with a radius of 4.1 cm, an arc of 18.3 cm subtends a central angle  . Find
the measure of  , correct to the nearest tenth of a degree.
(c) A pendulum swings through an angle of 450 . Find the length of the pendulum (to the
nearest tenth of a cm) if the end of the pendulum swings through an arc of length 32 cm.
6. Give the smallest possible positive and negative coterminal angle to each of the following.
17

(a) 2000 
(b) 2900 
(c)
6
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7. Write the equation of a circle with the given radius, and its centre at (0, 0).
a) 4 units
`b)
5 units
c)
9.1 units
d)
11 units
8. Which point(s) lies on the unit circle? Explain how you know.
(a)

12 5
,
13 13


(b)
1
1
,
2
2

(c)
  2 ,  2


 3

9. Each of the following points lies on the unit circle. Find the missing coordinate satisfying the
given conditions.
a)
 , y  in quadrant IV
2
5
b)
 x,  in quadrant III
5
6
10. The point P(x, y) is the point at the intersection of angle . If P() is the point at the
intersection of the terminal arm of angle  and the unit circle, determine the exact coordinates of
each.
a) P
 
7
4
b)
 
P 
3
2
 1
c)
11. Consider a point where P()    ,
 2
P()
d)
P
 
7
6
3
.
2 

a) Determine the coordinates of
P   .

2
b) Determine the coordinates of
P   .

2

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12. Given that the terminal arm of an angle of rotation,  , passes through the following points,
determine the exact values of the six trigonometric ratios for angle  and  .
b g
(a) P 9,40
(b) P
d3,3i
bg
13. (a) If the terminal arm of angle  , in standard position passes through point P a, b , then the
exact value of sec is
(b) If the terminal arm of angle  , in standard position passes through point P  b, 2b  , then
the exact value of sin  is
14. The following points are the intersection points between the unit circle and the terminal arm of an
angle in standard position. Determine the measure of the smallest positive angle:
(a) as a radian (exact value)
(b) as a degree
F 3 , 1I
G
H 2 2J
K
1 
 1
P
,

2
 2
P 
15. Determine the approximate value of the following ratios (to 2 decimal places where required).
(a) cos
3
4
(b) cot 4.47
(c) csc150
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16. Determine the exact value for each of the following.
(a) sin 3300
(d) cot
4
3

(b) tan1350
(c) cos 3150
3
(e) csc   
 4 
(f) sec

19
6
 3 
 2 
 
(g) 2sin    tan 2 
  cos   
 4 
 3 
 2
17. Determine  for 0    3600 . Round answers to the nearest whole degree.
(a) sin  0.6485
(b) cos  08219
.
(c) sec  10415
.
18. Determine  , 0    360o . Round any decimal answers to the nearest whole degree.
(a) 5sin  2  sin  1
19. Determine 
(a) cos  
(d) sin   0
(b) 2 sec  sec  3628
.
for 0    2 .
1
2
3
2
(b) tan   3
(c) sin  
(e) csc  2
(f) cot   1
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20. Solve the following trigonometric equations, using exact values if possible, or correct to the
nearest hundredth of a radian, on the domain 0    2
(a)
2sin 2   sin   0
(b)
cos 2   4cos   3  0
(c)
sec2   sec   2  0
(d)
tan 2   1  0
21. Solve the following trigonometric equations (algebraically), correct to the nearest degree, on
o
the domain 0o    360
(c) 3sec 2 x  4
(d)
8  2sin 2   10  5sin 
22. Find the general solutions for  , using exact values if possible, where  is in radians
(b) 2cos 2   5cos  3
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