1. THE UNIT CIRCLE
Reference Angles
And Trigonometry

2. Using Trigonometry in a Right Triangle
We were limited to Acute Angles
We can extend Trigonometry to
Angles of Any Measure
by placing those angles in
the coordinate plane
We do this by using reference angles,
Acute Angles measured to the x-axis.

3. The terminal side
is rotated counter-clockwise.
135°
Angles are Placed with one side
called the initial side
on the positive x-axis.

4. The terminal side
is rotated counter-clockwise.
135°
45°
A Reference Angle is measured
to the x-axis.

5. The terminal side
is rotated counter-clockwise.
225°
45°
A Reference Angle is measured
to the x-axis.

6. The terminal side
is rotated counter-clockwise.
315°
45°
A Reference Angle is measured
to the x-axis.

7. If the terminal side is rotated clockwise, the angle measure is
Negative.
45°
A Reference Angle is measured
to the x-axis.
Always Positive.
-45°

8. Unit Circle
has a radius
of 1 unit.

9. Unit Circle has a radius of 1 unit.
Cos +
Sin + =y
45° x=
Cosine = x
Sine = y 1
45°

10. Cos -
Sin +
Cos +
Sin +
135°
Reference Angle =
45°
45°

11. 135° 45° 225° 45° 45° 45° Reference Angle = 225° Cos - Sin + Cos +

12. 135° 45° 45° 45° 45° 45° Reference Angle 315° 225° 315° Cos - Sin +

13. 135° 45° 45° 45° 45° 45° 225° 315° Quadrant 2 Cos - Sin + Quadrant 1

14. Cosine = x Sine = y Tangent = Δy Δx Tangent = Sine Cosine tan = 1 45°
Quadrant 1
Cos +
Sin +
tan = 1
Cosine = x
Sine = y
45°
45°
Tangent = Δy Δx
Tangent = Sine
Cosine

15. tan = -1 tan = 1 135° 45° 45° 45° 45° 45° 225° 315° tan = 1 tan = -1
Quadrant 2
Cos -
Sin +
Quadrant 1
tan = -1
Cos +
Sin +
tan = 1
135°
45°
45°
45°
45°
45°
225°
315°
Quadrant 3
Cos -
Sin -
Cos +
Sin -
Quadrant 4
tan = 1
tan = -1

16. Tangent = Sine Cosine Quadrant 2 Cos - Sin + Tan - Quadrant 1 Cos +

17. Cos +
Sin +
30°
30° 1
Cosine = x
Sine = y

18. 150° 150° 30° 30° 30° 30° 30° 210° 330° Cos - Sin + Cos + Sin + Cos -

19. Tangent = Sine Cosine 150° 30° 210° 330° Cos - Sin + Cos + Sin + Cos -

20. Cos +
Sin +
60°
60° 1

21. Cos +
Sin +
Cos +
Sin +
60°
120°
120°
60°
60°

22. Cos -
Sin +
Cos +
Sin +
60°
120°
60°
60°
60°
Cos -
Sin -
240°

23. 60° 120° 60° 60° 60° 60° 240° 300° Cos - Sin + Cos + Sin + Cos - Sin -

24. Tangent = Sine Cosine 60° 120° 240° 300° Cos - Sin + Cos + Sin + Cos -

25. Cosine = x Sine = y Cos(180) = -1 Cos(90) =0 Sin(180) = 0 Sin(90) = 1
90°
(0 , 1)
180°
Cos -
Sin
Cosine = x
Sine = y
Cos +
Sin
(-1 , 0)
(1 , 0)
Cos(0) =1
Sin(0) = 0
270°
Cos(270) = 0
Sin(270) = -1
Cos
Sin -
(0 , -1)

26. Cosine = x Sine = y Tangent = Sine Cosine Cos(180) = -1 Sin(180) = 0
90°
Tan(180) =0
(0 , 1)
Tan(90)
undefined
180°
Cos -
Sin
Cosine = x
Sine = y
Cos +
Sin
(-1 , 0)
(1 , 0)
Tangent = Sine
Cosine
Cos(0) =1
Sin(0) = 0
270°
Tan(0) =0
Cos(270) = 0
Sin(270) = -1
Cos
Sin -
Tan(270)
undefined
(0 , -1)

27. Evaluate the trigonometric functions at each real number.
= y
= x

28. Evaluate the six trigonometric functions at each real number.
(0, -1)
= y
= -1
= -1
= x
= 0
DNE
= 0
DNE
Does Not Exist

29. Evaluate the six trigonometric functions at each real number.-1
Sin
Cos
Tan
Csc
Sec
Cot -1
So, you think you got it now?