1. 12-3 Trigonometric Functions of General Angles
Part 1: Points and Reference Angles

2. Trig Functions for Any Angle
Previously we have defined trig functions for acute angles only; now we will use angles of any measure (in standard position).
For any angle θ and any point (x, y) on the terminal side and let r be the distance from the origin to (x, y): y y y
(x, y) r θ θ θ x x x r
(x, y) r
(x, y)

3. Finding Trig Functions Using a Point
Find the values of the six trig functions of an angle θ in standard position whose terminal side passes through (8, -15).

4. When an angle θ is in standard position its terminal side passes through (-3, -4). Find the values of the six trig functions of θ.

5. Signs of Functions
In the previous definitions, r will always be positive, so the sign of the function will be determined by the sign of x or y, which depend on the quadrant of the terminal side of θ.
*sine and cosine are defined for all angles, but the other four are undefined for certain quadrantal angles
Function
value
Quadrant of θ
sin θ
csc θ + -
cos θ
sec θ
tan θ
cot θ

6. Determining Defined Functions
Determine which functions are defined for a 180 angle and find their values.

7. Determine which functions are defined for a 90 angle and find their values.

8. Reference Angles
If θ is not a quadrantal angle, there is a unique acute angle α, corresponding to θ, formed by the terminal side of θ and the x-axis (positive or negative, depending on the location of the terminal side).
When θ is in standard position, α is called the reference angle of θ. θ θ α α α α θ θ
θ = -210
α = 30
θ = 420
α = 60
θ = 300
α = 60
θ = 225
α = 45

9. Finding Reference Angles
Find the measure of the reference angle α for each given angle θ.
θ = 140
θ = 30010’
θ = -135
How would you find α for a positive angle in quadrant I? Negative?

10. For each angle θ, find its reference angle.
θ = 170
θ = 250
θ = -305

11. Functions of Acute Angles
Write cos 200 as a function of an acute angle.

12. Write sin 130 as a function of an acute angle.