Radian Measures 3/06

Topic: Periodic Functions and Trigonometry
Aim: How can we find exact values of cosine and sine using radian measures?
Do Now: What is the exact value of the cos π/4 radians?
Finding cos π/4 radians
π/4 radians = ¼ of a straight angle because ¼ x π = π/4. ¼ x 180 = 45.

π radians = 180° (straight angle)

Also, π/4 radians x 180/π radians = 45°

Cos π/4 = cos 45 = √2/2.
Finding cosine and sine of a radian measure
What are the exact values of cos (7π/6 radians) and sin (7π/6 radians)?
7π/6 radians x 180/π radians =

7 x 30 = 210°

Reference angle = 60° and in Quad III, where cos and sin are both negative (-)

Cos (7π/6 radians) = -  √3/2
Sin (7π/6 radians ) = - ½

Practice 13-3
Complete problems #1 – 20  ALL.
Exit slip
How many radians are in  60°?


How many degrees are in 5π/6 radians?


Sketch these angles in the unit circle and label their x and y coordinates.
Essential Understanding
An angle with a full circle rotation measures 2π radians.
An angle with a semicircle rotation measures π radians.

Recall: π = 3.14159……. (is this a rational or irrational number?)
Vocabulary (visualized)
Key Concept
Proportion relating radians and degrees

d°/180° = r radians/π radians

Use this proportion to convert between radians and degrees.
Why does the proportion work?
Recall that the circumference of a circle is 2πr, so there are 2π radians in any circle.

Since 2π radians = 360°, then π radians = 180°.

To convert degrees to radians, multiply by π radians/180°
To convert radians to degrees, multiply by 180°/π radians.
Vocabulary
Central Angle: an angle of a circle with vertex at the center.
Intercepted arc: portion of the circle with endpoints on the sides of the central angle and remaining points within the interior of the angle
Radian: measure of a central angle that intercepts an arc with length equal to the radius of the circle.
Using conversion factors/dimensional analysis
Problem 1:

What is the degree measure of an angle of -3π/4 radians?
Solution:
-3π/4 radians = -3π/4 radians X 180°/π radians

An angle of -3π/4 radians = -135°

Problem 2
What is the radian measure of an angle of 27°?

Solution:

27° = 27° X π radians/180° = 3π/20 radians.
An angle of 27° measures 3π/20 radians.
Complete the 4 questions on handout page 837 (#1 a-d)
a. π/2 radians = _____ °

b. 225° = ______ radians

c. 2 radians = _____ °

d. 150° = _____ radians