Download 9.  10.  SOLUTION:  

terminal side of the angle because r = 1.
4-3 Trigonometric Functions on the Unit Circle
Find the exact value of each trigonometric
function, if defined. If not defined, write
undefined.
9. sin
13. cos (–270°)
SOLUTION: The terminal side of
in standard position lies on the positive y-axis. Choose a point P(0, 1) on the
terminal side of the angle because r = 1.
SOLUTION: The terminal side of in standard position lies on the positive y-axis. Choose a point P(0, 1) on the
terminal side of the angle because r = 1.
14. sec 180°
SOLUTION: The terminal side of
in standard position lies on
the negative x-axis. Choose a point P( , 0) on the
terminal side of the angle because r = 1.
10. tan 2π
SOLUTION: The terminal side of
in standard position lies on the positive x-axis. Choose a point P(1, 0) on the
terminal side of the angle because r = 1.
15. tan π
SOLUTION: The terminal side of π in standard position lies on the
negative x-axis. Choose a point P( , 0) on the
terminal side of the angle because r = 1.
11. cot (–180°)
SOLUTION: The terminal side of
in standard position lies on the negative x-axis. Choose a point P( , 0) on
the terminal side of the angle because r = 1.
16. SOLUTION: 12. csc 270°
The terminal side of −
in standard position lies on
SOLUTION: The terminal side of
in standard position lies on
the negative y-axis. Choose a point P(0, ) on the
terminal side of the angle because r = 1.
13. cos (–270°)
SOLUTION: eSolutions Manual - Powered by Cognero
The terminal side of
in standard position lies on the positive y-axis. Choose a point P(0, 1) on the
terminal side of the angle because r = 1.
the negative y-axis. Choose a point P(0,
terminal side of the angle because r = 1.
) on the
Sketch each angle. Then find its reference
angle.
17. 210°
SOLUTION: Page 1
The terminal side of 210º lies in Quadrant III.
Therefore, its reference angle is θ ' = 210º – 180º or
4-3 Trigonometric Functions on the Unit Circle
Sketch each angle. Then find its reference
angle.
17. 210°
19. SOLUTION: SOLUTION: The terminal side of
lies in Quadrant II. Therefore, its reference angle is θ ' =
The terminal side of 210º lies in Quadrant III.
Therefore, its reference angle is θ ' = 210º – 180º or
30º.
.
20. 18. 135°
SOLUTION: The terminal side of 135º lies in Quadrant II.
Therefore, its reference angle is θ ' = 180º – 135º or
45º.
SOLUTION: A coterminal angle is
− 2π or
, which lies
in Quadrant IV. So, the reference angle is θ ' is 2π −
or
.
19. SOLUTION: The terminal side of
lies in Quadrant II. Therefore, its reference angle is θ ' =
.
eSolutions Manual - Powered by Cognero
21. −405°
SOLUTION: A coterminal angle is −405° + 360(2)° or 315°. The terminal side of 315° lies in Quadrant IV, so its
reference angle is 360º – 315º or 45º. Page 2
4-3 Trigonometric Functions on the Unit Circle
21. −405°
24. SOLUTION: A coterminal angle is −405° + 360(2)° or 315°. The terminal side of 315° lies in Quadrant IV, so its
reference angle is 360º – 315º or 45º. SOLUTION: A coterminal angle is
terminal side of
reference angle is
The + 2(−1)π or
lies in Quadrant I, so the
22. −75°
SOLUTION: A coterminal angle is −75° + 360° or 285°. The
terminal side of 285° lies in Quadrant IV, so its
reference angle is 360° − 285° or 75°.
Find the exact value of each expression.
25. cos
SOLUTION: Because the terminal side of θ lies in Quadrant III,
the reference angle θ ' is – π or
. 23. SOLUTION: The terminal side of lies in Quadrant II. In Quadrant III, cos θ is negative and
Therefore, its reference angle is θ ' =
.
26. tan
SOLUTION: Because the terminal side of θ lies in Quadrant III,
the reference angle θ ' is 24. eSolutions
Manual - Powered by Cognero
SOLUTION: A coterminal angle is
+ 2(−1)π or
or . Page 3
The 4-3 Trigonometric Functions on the Unit Circle
28. cot (−45°)
26. tan
SOLUTION: SOLUTION: Because the terminal side of θ lies in Quadrant III,
the reference angle θ ' is or . A coterminal angle is −45° + 360° or 315°. Because the terminal side of 315° lies in Quadrant IV, the
reference angle θ ' is 360° − 315° or 45°. Because tangent and cotangent are reciprocal functions and
tan θ is negative in Quadrant IV, it follows that cot θ
is also negative in Quadrant IV.
.
In Quadrant III, tan θ is positive and
27. sin
SOLUTION: Because the terminal side of θ lies in Quadrant II,
the reference angle θ ' is or 29. csc 390°
.
SOLUTION: In Quadrant II, sin θ is positive and
.
A coterminal angle is 390° + 360° or 30°, which lies in Quadrant I. So, the reference angle θ ' is 360° −
30° or 30°. Because sine and cosecant are reciprocal functions and sin θ is positive in Quadrant
I, it follows that csc θ is also positive in Quadrant I.
28. cot (−45°)
SOLUTION: A coterminal angle is −45° + 360° or 315°. Because the terminal side of 315° lies in Quadrant IV, the
reference angle θ ' is 360° − 315° or 45°. Because tangent and cotangent are reciprocal functions and
eSolutions
- Powered
Cognero IV, it follows that cot θ
tan θManual
is negative
in by
Quadrant
is also negative in Quadrant IV.
Page 4
30. sec (−150°)
4-3 Trigonometric Functions on the Unit Circle
29. csc 390°
30. sec (−150°)
SOLUTION: SOLUTION: A coterminal angle is 390° + 360° or 30°, which lies in Quadrant I. So, the reference angle θ ' is 360° −
30° or 30°. Because sine and cosecant are reciprocal functions and sin θ is positive in Quadrant
I, it follows that csc θ is also positive in Quadrant I.
A coterminal angle is −150° + 360° or 210°, which lies in Quadrant III.
Because the terminal side of θ lies in Quadrant III.
So, the reference angle θ ' is 210º – 180º or 30º. Because secant and cosine are reciprocal functions
and cos θ is negative in Quadrant III, it follows that
sec θ is also negative in Quadrant III.
30. sec (−150°)
SOLUTION: A coterminal angle is −150° + 360° or 210°, which lies in Quadrant III.
Because the terminal side of θ lies in Quadrant III.
So, the reference angle θ ' is 210º – 180º or 30º. 31. tan
SOLUTION: Because the terminal side of θ lies in Quadrant IV,
the reference angle θ ' is Because secant and cosine are reciprocal functions
and cos θ is negative in Quadrant III, it follows that
sec θ is also negative in Quadrant III.
Quadrant IV, tan θ is negative.
or . In
eSolutions Manual - Powered by Cognero
Page 5
4-3 Trigonometric Functions on the Unit Circle
32. sin 300°
31. tan
SOLUTION: SOLUTION: Because the terminal side of θ lies in Quadrant IV,
the reference angle θ ' is or Because the terminal side of θ lies in Quadrant IV,
the reference angle θ ' is .
or . In
Quadrant IV, tan θ is negative.
In Quadrant IV, sin θ is negative.
32. sin 300°
SOLUTION: Because the terminal side of θ lies in Quadrant IV,
the reference angle θ ' is .
or In Quadrant IV, sin θ is negative.
eSolutions Manual - Powered by Cognero
Page 6