Chapter 1.6 Trigonometric Functions

The Unit Circle

Degree/Radian Conversion To convert a degree measure to radians, multiply by π radians180°

To convert a radian measure to degrees, multiply by 180°π radians

Examples Examples

1) 120°

2) -45°

3) 5π6

4) -3π2

Radian Measure The RADIAN MEASURE of the angle ACB at the center of the unit circle

equals the length of the arc that ACB cuts from the unit circle. Radius =1

Finding Arc Length Find the length of an arc on a circle of radius 3 by a central angle of measure

2π/3.

S = r θ

= 3(2π/3)

= 2π

An Angle θ In Standard Position

When an angle of measure θ is placed in standard position at the center of a circle of radius r, the six trigonometric functions of θ are defined as follows:

sin θ = y/r csc θ = r/y

Cos θ = x/r sec θ = r/x

Tan θ = y/x cot θ = x/y

(SOHCAHTOA) Sin – opp/hyp

Cos – adj/hyp

Tan – opp/adj

Csc – hyp/opp

Sec – hyp/adj

Cot – adj/opp

Graph of sin

Graph of cos

Graph of tan

Periodicity Periodic Function, Period: A function f(x) is periodic if there is a postive

number p such that f(x + p) = f(x) for every value of x. The smallest such value of p is the period of f.

Transformations of Trigonometric Graphs Y = a f ( b ( x + c ) ) + d

A = vertical stretch or shrink/reflection about x-axis

B = horizontal stretch or shrink/ reflection about y-axis

C = Horizontal shift

D = vertical shift

Finding Angles in degrees and Radians Find the measure of cos-1 (-0.5) in degrees and radians.

Put the calculator in degree mode and enter cos-1 (-0.5). You will get 120 degrees.

Using the Inverse Trigonometric Functions Sinx = 0.7

Take the sin-1 of both sides.

X = sin-1(0.7)

X = 0.775

Homework Quick Review pg 52 # 1-4

Section 1.6 Exercises pg 52 #1-10