Download Geometry H Additional Topics in Trigonometry 3. Reference Angles

Geometry H
Additional Topics in Trigonometry
3. Reference Angles
Finding a reference angle
A reference angle is an angle whose
measure is between 0° and 90°
(not including 0° or 90°).
Every angle in Quad II, III, and IV has a
reference angle in Quad I. In many
ways an angle behaves the same way as
its reference angle. Therefore, a small
familiar angle can do much of the work
of a large angle. In addition, your
calculator and trig tables deliver their
information in terms of reference angles.
How an angle’s reference angle is
computed depends on what quadrant the
angle terminates in. Here is a summary.
(All are provable by congruent
triangles.)
Find the reference angle for 600°
First, using co-terminal angles, express
the angle in terms of an angle between
0° and 360°. 600° - 360° = 240°.
240° is in Quad III, so the reference
angle is 240° - 180° = 60°.
Find the reference angle for -265°.
First, using co-terminal angles, express
the angle in terms of an angle between
0° and 360°. -265° + 360° = 95°.
95° is in Quad II, so the reference angle
is 180° - 95° = 85°.
Find the reference angle for 790°.
First, using co-terminal angles, express
the angle in terms of an angle between
0° and 360°. 790° - 360° - 360° = 70°.
70° is between 0° and 90°.
If θ is your angle, here’s how to find its
reference angle.
Expressing trigonometric functions in
terms of a reference angles.
Quadrant
II
III
The sine, cosine, and tangent of an angle
are the same as the sine, cosine, and
tangent of its reference angle except
when it comes to being positive or
negative.
IV
Reference Angle

180° - θ

θ - 180°

360° - θ
Examples
Find the reference angle for 140°
140° is in Quad II, so the reference angle
is 180° - 140° = 40°
Find the reference angle for 237°
237° is in Quad III, so the reference
angle is 237° - 180° = 57°
Find the reference angle for 320.6°
320.6° is in Quad IV, so the reference
angle is 360° - 320.6° = 39.4°
We have said that a trig function is
positive or negative depending on the
quadrant. Sine 200° is negative because
sine is negative in Quad III. Its
reference angle is 20°. (200° - 180°)
So, sin 200° = -sin20°.
Examples
Express cos 130° in terms of a
reference angle.
Cosine is negative in Quad II, and the
reference angle is 50°. (180° - 130°)
So, cos 130° = -cos50°.
Geometry H
Additional Topics in Trigonometry
3. Reference Angles
Express tan 212° in terms of a
reference angle.
Tangent is positive in Quad III, and the
reference angle is 32°. (212° - 180°)
So, cos 130° = -cos50°.
Express sin 523.2° in terms of a
reference angle.
First, using co-terminal angles, express
the angle in terms of an angle between
0° and 360°. 523.2° - 360° = 163.2°.
Sine is positive in Quad II, and the
reference angle is 16.8°. (180° - 163.2°)
So, sin 523.2 = sin 16.8°.
Express tan( -70°) in terms of a
reference angle.
First, using co-terminal angles, express
the angle in terms of an angle between
0° and 360°. -70° + 360° = 290°.
Tangent is negative in Quad IV, and the
reference angle is 70°. (360° - 290°)
So, tan(-70°) = -tan 70°.
16. 901°
17. 822°
18. 750°
19. -121°
20. -229°
21. -325°
22. –13.5
23. –104.6° 24. –254.3
Express the trig function in terms of a
reference angle.
25. sin 101° 26. cos 101° 27. tan 101°
28. sin 224° 29. cos 224° 30. tan 224°
31. sin 314° 32. cos 314° 33. tan 314°
34. sin 195.1° 35. cos 195.1° 36. tan 195.1°
37. sin 284.7° 38. cos 284.7° 39. tan 284.7°
Exercises
Give the angle’s reference angle.
1. 145°
2. 216°
3. 303°
40. sin 467° 41. cos 467° 42. tan 467°
43. sin 695° 44. cos 695° 45. tan 695°
46. sin(-35°) 47. cos(-35°) 48. tan(-35°)
4. 295°
5. 98°
6. 198°
49. sin(-113°) 50. cos(-113°) 51. tan(-113°)
7. 102.4°
8. 345.6°
9. 247.2°
10. 213.5
11. 165.9° 12. 355.1°
13. 657°
14. 457°
15. 503°