5.5 Circular Functions: Graphs and PropertiesMon Nov 10

Do NowEvaluate

1) Sin pi/22) Cos 2pi3) Tan pi/4

Quiz Review

• Retakes by next Wed

Circular Functions

• The domains of the trigonometric functions have been sets of angles or rotations.

• Trigonometric functions with domains composed of real numbers (radians) are called circular functions

Basic Circular Functions

• With respect to S radians and the unit circle,

Reflections on the Unit Circle

• Because the unit circle is symmetric, we can use the coordinates of one point on the unit circle to find coordinates of its reflections

• (Draw unit circle with point x,y )

Ex1

• The point (3/5, 4/5) lies on the unit circle. Find its reflections across the x-axis, y-axis, and origin

Finding Function Values

• Knowing the coordinates of only a few points on the unit circle + reflections allows you to find many trig values

Ex

• Find each of the following using reflections• 1)

• 2)

• 3)

Calculator and Radians

• When working in radians, make sure the calculator is set to radians mode!

• MODE -> RADIANS should be highlighted

Graphs of Sine and Cosine Functions

• One characteristic about circular functions is that they repeat, or oscillate

• Since circular functions travel on a circle, they tend to repeat every revolution

Graph of y = sin x

• Let’s look at the y-coordinate on the unit circle

Graph of y = cos x

• Let’s look at the x-coordinate of the unit circle

Domain and Range

• The domain and range of both sine and cosine functions are the same

• Domain = All real numbers• Range = [-1, 1]

Period of sine and cosine

• A function with a repeating pattern is called periodic. All trig graphs are periodic.

• The period of y = sin x and y = cos x is 2pi

Amplitude

• The amplitude of a periodic function is defined as one half of the distance between its maximum and minimum values

• You can calculate this by dividing the range by 2

• The amplitude of y = sin x and y = cos x is 1

Symmetry

• Sin (-x) = - sin x• The sine function is odd

• Cos (-x) = cos x• The cosine function is even

Closure

• Describe the graphs of the sine and cosine functions. How are they similar? Different?

• HW: p.505 #1-43 odds

5.5 Sine and Cosine Graph ReviewTues Nov 11

• Do Now• Evaluate• 1)

• 2)

HW Review: p.505 #1-47

Review of Sine and Cosine Graph

(If time) The other 4 trig graphs

• Tan x and cot x

• These 2 graphs have asymptotes

Sec x and csc x

• These 2 also have asymptotes

Closure

• Why are the other 4 trig graphs different than sine and cosine?