1. Trigonometric Functions: Unit Circle Approach

2. Radians vs. Degrees
Measurements of common angles

3. Unit Circle Unit circle: Circle with radius 1 centered at the origin
Equation: x2 + y2 = 1
Circumference: 2π

4. Unit Circle
Travel t units around circle, starting from the point (1,0), ending at the point P = (x, y). Remember radians is the arc length in number of radii travelled as an angle rotates.
The point P = (x, y) is used to define the trigonometric functions of t.

5. Trigonometric Functions
Let t be a real number and P = (x, y) the point on the unit circle corresponding to t:
Sine function: y-coordinate of P
sin t = y
Cosine function: x-coordinate of P
cos t = x
Tangent function: if x  0

6. Trigonometric Functions
Let t be a real number and P = (x, y) the point on the unit circle corresponding to t:
Cosecant function: if y  0
Secant function: if x  0
Cotangent function: if y  0

7. Signs of the Trigonometric FunctionsII
III IV
ALL
STUDENTS
TAKE
CALCULUS

8. Exact Values Using Points on the Circle
A point on the unit circle will satisfy the equation x2 + y2 = 1
Use this information together with the definitions of the trigonometric functions.

9. Exact Values Using Points on the Circle
Example. Let t be a real number and P = the point on the unit circle that corresponds to t.
Problem: Find the values of sin t, cos t, tan t, csc t, sec t and cot t

10. Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of θ
Problem: θ = = 90°

11. Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of θ
Problem: θ = π = 180°

12. Exact Values for Quadrantal Angles
Example. Find the values of the trigonometric functions of θ
Problem: θ = = 270°

13. Exact Values for Quadrantal Angles

14. Exact Values for Quadrantal Angles
Example. Find the exact values of
Problem: sin(90°)
(b) Problem: cos(5π)

15. Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of θ
Problem: θ = = 45°

16. Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of θ
Problem: θ = = 60°

17. Exact Values for Standard Angles
Example. Find the values of the trigonometric functions of θ
Problem: θ = = 30°

18. Exact Values for Standard Angles

19. Exact Values for Standard Angles
Example. Find the values of the following expressions
(a) Problem: sin(315°)
(b) Problem: cos(120°)
(c) Problem:

20. Circles of Radius r Theorem.
For an angle θ in standard position, let P = (x, y) be the point on the terminal side of θ that is also on the circle x2 + y2 = r2. Then

21. Circles of Radius r Example.
Problem: Find the exact values of each of the trigonometric functions of an angle µ if (-12, -5) is a point on its terminal side.