1. Trigonometry Extended: The Circular Functions
4.3
Trigonometry Extended: The Circular Functions

2. Warm-up 1)Show work. Convert to degrees Change to Radians 2) 3)
2) 3)
Change to degrees
4) 5)
Find the arc length, given  and r.
6)  = /2; r = 2 7)  = 45 r = 6

3. Right triangle trig

4. Finding the trig functions with one trig ratio
Given:

5. Your turn
Given:

6. Homework “Mini-Quiz” Construct the special angles chart.
What are the six trig. function values for the  below?
Find the six trig. function values of 60. c a  b

7. Trig extended: the circular functions
Terminal side
KEY TERMS:
Initial side-
Terminal side-
Vertex-
Measure of angle-
Initial side
The ray where an angle begins
The ray where an angle ends
Point where the initial and terminal rays meet
A number that describes the amount of rotation from the initial side to the terminal side of an angle.

8. More terms Standard position- Positive angles- Negative angles-y
More terms
135° x
-225°
When the initial ray is the positive side of the x-axis
Standard position-
Positive angles-
Negative angles-
Coterminal angles-
measured counterclockwise from the initial ray
measured clockwise from the initial ray
Angles with the same terminal rays

9. Circular definition of trig. functionsy
Draw the positive angle in standard position which contains the point (x,y). Label it .
What does the segment from the origin to the point represent on the circle?
What side of the triangle does the x-coordinate represent? The y-coordinate?
Express the six trig. ratios in terms of the point (x,y) and the radius of the circle, r.
(x,y) r y  x x

10. Evaluating an angle in the Cartesian plane
Where an angle is in standard position and (x,y) is any point on the terminal side (except the origin) and r is the distance from the origin to the point (x,y).

11. All Students Take Calculus
Quadrants in which the three basic trig. functions are positive (and their reciprocals as well).
Sin All
Tan Cos

12. Reference angle
When any angle is in the standard position, we can use a reference triangle, to help us find the trig. functions.
Example: (-2,3)

13. You try.
Find a point on the terminal side of 315, and then find the six trig functions of a 315. (Assume angle is in standard position)
Review Question: What is 315 in radians?