1. Trigonometric Functions of Any Angles
Objective:
To evaluate trig functions of any angle by using reference angles

2. (x, y) θ r y r
sin θ =
csc θ = r y x r
cos θ =
sec θ = r x y x
tan θ =
cot θ = x y

3. Evaluate the six trigonometric functions if the point (-4,3) lies on the terminal side of an angle θ. 3 5
sin θ =
csc θ = 5 3 -4 5
cos θ =
sec θ = 5 -4 3 -4
tan θ =
cot θ = -4 3

4. Evaluate the six trigonometric functions if the point (1,-3) lies on the terminal side of an angle θ.
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =

5. Evaluate the six trigonometric functions if the point (-7,-4) lies on the terminal side of an angle θ.
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =

6. Evaluate the six trigonometric functions if the point (3, 0) lies on the terminal side of an angle θ.
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =

7. When Trig Functions are +/-
sin, csc +
cos, sec –
tan, cot –
sin, csc +
cos, sec +
tan, cot + II I
(-,+)
(+,+)
III IV
sin, csc –
cos, sec –
tan, cot +
sin, csc –
cos, sec +
tan, cot –
(-,-)
(+,-)

8. Name the quadrant in which the angle lies.
sinθ < 0 and cosθ < 0
sinθ > 0 and tanθ < 0
III II

9. Evaluate the six trigonometric functions if the point (0, 3) lies on the terminal side of an angle θ.
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =

10. Evaluate the six trigonometric functions if the point (-3, 0) lies on the terminal side of an angle θ.
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =

11. Evaluate the six trigonometric functions if the point (0, -3) lies on the terminal side of an angle θ.
sin θ =
csc θ =
cos θ =
sec θ =
tan θ =
cot θ =

12. If (3,0) is on the terminal side of the angle, then θ = 0°.
90°
(0,3)
(3,0)
If (3,0) is on the terminal side of the angle, then θ = 0°.
If (0,3) is on the terminal side of the angle, then θ = 90°

13. If (-3,0) is on the terminal side of the angle, then θ = 180°
270°
(-3,0)
(0,-3)
If (-3,0) is on the terminal side of the angle, then θ = 180°
If (0,-3) is on the terminal side of the angle, then θ = 270°

14. Quadrantal Angle Angle whose terminal side falls on an axis.
Examples: 0°, 90°, 180°, 270°, 360°

15. Trig Functions of Quadrantal Anglesθ
sin
cos
tan
csc
sec
cot 0°
90°
180°
270°
360°

16. Reference Angles Remember: Must be positive Must be acute
The acute angle formed by the terminal side of the angle and the horizontal axis (x-axis).
Remember:
Must be positive
Must be acute

17. Find the reference angle.
140°
180° – 140°
= 40°

18. Find the reference angle.
230°
230° – 180°
= 50°

19. Short-Cuts for Reference Angles
Quadrant
θ in degrees
θ in radians I II
III IV
If θ goes around more than once… θ θ
180° - θ
π - θ
θ - 180°
θ - π
360° - θ
2π - θ
θ - 360°.
Then proceed with above.
θ - 2π
Then proceed with above.

21. Find the reference angle.
310
170
305
360° - 310°
= 50°
180° °
= 10°
360° - 305°
= 55°

22. Find the reference angle.

23. Evaluate the following.
225° - 180°
= 45°
cos(45°)
Quadrant III
In Quadrant III, cos is negative.
180° - 120°
= 60°
Quadrant II
tan(60°)
In Quadrant II, tan is negative.

24. Evaluate the following.
Quadrant III
In Quadrant III, tan is positive.

25. Evaluate the following.
Quadrant III
In Quadrant III, sin is negative.

26. Find the reference angle.
375
470
595
375° - 360°
= 15°
470° °
= 110°
180° °
= 70°
595° - 360°
= 235°
235° °
= 55°

27. Evaluate the following.
It’s larger than 360°!
390° - 360°
= 30°
Quadrant I.
sin(30°)
In Quadrant I, sin is positive.

28. Evaluate the following.
It’s larger than 2π!
Quadrant I!

29. If θ is negative… Add 360° (or 2π) until you get a positive angle.
Proceed as usual.

30. Find the reference angle.
-275
-190
-112
-275° + 360°
= 85°
-190° °
= 170°
180° - 170°
= 10°
-112° + 360°
= 248°
248° °
= 68°

31. Find the reference angle.
Quadrant III.

32. Evaluate the following.
It’s negative.
= 240°
Quadrant III.
240° - 180°
= 60°
sin(60°)
In Quadrant III, sin is negative.