1. Do Now
A central angle of a circle with radius 150 cm cuts off an arc of 200 cm. Find each measure:
the radian measure of the angle.
the area of a sector with that central angle.

2. The Unit Circle and Circular Functions
3.3
The Unit Circle and Circular Functions
Objective: SWBAT use the unit circle to determine circular functions.

3. Circular Functions
A unit circle has its center at the origin and a radius of 1 unit.
The trigonometric functions of angle θ in radians are found by choosing a point (x, y) on the unit circle can be rewritten as functions of the arc length s.
When interpreted this way, they are called circular functions.

4. Circular Functions
For any real number s represented by a directed arc on the unit circle,
Since we use the Unit Circle we can replace θ with s (arc length) because r = 1!

5. The Unit Circle

6. Evaluating A Circular Function
Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians.
This applies both to methods of finding exact values (such as reference angle analysis) and to calculator approximations.
Calculators must be in radian mode when finding circular function values.

7. FINDING EXACT CIRCULAR FUNCTION VALUES
Find the exact values of:
Evaluating a circular function at the real number is equivalent to evaluating it at radians.
An angle of intersects the circle at the point (0, –1).
Since sin s = y, cos s = x, and

8. FINDING EXACT CIRCULAR FUNCTION VALUES
Use the figure to find the exact values of
The real number
corresponds to the unit circle point

9. FINDING EXACT CIRCULAR FUNCTION VALUES
Use the figure and the definition of tangent to find the exact value of
negative direction
yields the same ending point as moving around the
Moving around the unit circle units in the
circle units in the positive direction.

10. FINDING EXACT CIRCULAR FUNCTION VALUES Cntd.
corresponds to

11. FINDING EXACT CIRCULAR FUNCTION VALUES
Use reference angles and degree/radian or your unit circle conversion to find the exact value of

12. Caution
A common error in trigonometry is using a calculator in degree mode when radian mode should be used.
Remember, if you are finding a circular function value of a real number, the calculator must be in radian mode.

13. APPROXIMATING CIRCULAR FUNCTION VALUES
Find a calculator approximation for each circular function value.
(c) cot
≈ .2552
(d) sec –2.9234
≈ –1.0243