This document provides an introduction to radian measure and reciprocal trigonometric functions. It defines a radian as the central angle that intercepts an arc of a circle equal in length to the radius. Examples are provided for converting between degrees and radians. The six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) are defined, with the last three being the reciprocals of the first three. Examples are given of finding all six trig functions given values of tangent or cosecant.

1. Intro To Trigonometry Day 4
Radian Measure/Reciprocal Functions
1. Name the quadrant in which an angle of measure x
could lie when:
a) sin x < 0 and cos x > 0
b) tan x > 0 and cos x < 0

2. A Central Angle of a circle is an angle with a vertex at the center of a circle.
Intercepted Arc
Central Angle
When a central angle intercepts an arc that has the same length as a
radius of the circle, the measure of the angle is defined to be one radian.
r
1 radian
r

3. The Radian Measure of an angle θ is:
arc length
s
θ = r
radius
Example: What is the radian measure of a half a revolution?
Example: What is the radian measure of the following?
a) 3600 = c) 900 =
b) 1800 = d) 300 =

4. Examples:
1. A circle with radius of 22 inches, has a central angle that intercepts an arc of exactly 22 inches.
What is the measure of this central angle (in radians)?
2. Example: A central angle of 3 radians in a circle intercepts an arc of 61 cm. Find the measure of
the radius to the nearest tenth.

5. Conversion Formulas
1. Degrees to Radians
multiply by π radians
0
180
2. Radians to Degrees
1800
multiply by π radians
Examples: Convert from degrees to radians.
a) 3150 b) ­1500
Examples: Convert from radians to degrees .
a) 2π b) π
3 4

6. Reciprocal Trigonometric Functions
In earlier lessons in this chapter, you studied three trigonometric functions—sine, cosine, and tangent.
Three other functions—cosecant, secant, and cotangent—are reciprocals of the three you have used.

7. Example: Find all 6 trig functions if:
a) tan θ = (√7/3) and sinθ<0

8. Example: Find all 6 trig functions if:
a) csc θ = (­√10/3) and cotθ<0

9. Attachments
Terminal­Initial Sides of Rotations.gsp