Precalculus

Section 7.5

Warmup

Graph the function. State the Domain, Range, Asymptotes, and Period

1. f(x) = -2 tan(1/3 x)2. f(x) = sec(2x) + 1

Warmup Answers

1. f(x) = -2 tan(1/3 x)Domain:

Range:

Asymptotes:

Period:

Warmup Answers

1. f(x) = sec(2x) + 1Domain:

Range:

Asymptotes:

Period:

7.5 Lesson – Unit Circle and Properties of Trig Functions

• You do not need to write down the information on this slide in your notes

• We have actually already done most of this section: we have talked about the unit circle and we discussed domain, range, and period while graphing the trig functions

• Today we will be adding one property: odd-even properties

7.5 Lesson – Unit Circle and Properties of Trig Functions

• You do not need to attempt to copy the following graphs

• Look for SYMMETRY in the graphs– Could the function be reflected over a line or a

point?– Example: Reflected over y-axis or reflected over

origin

What is the graph of f(x) = x ?

What is the graph of f(x) = x2 ?

What is the graph of f(x) = x3 ?

What is the graph of f(x) = x4 ?

What is the graph of f(x) = x5 ?

What is the graph of f(x) = x6 ?

Do you see the pattern?

• Odd powers: • Even powers:

(Write this down)

• An “odd” function reflects over the origin• An “even” function reflects over the y-axis

Function Notation Definitions of odd and even functions

• Odd function:f(-x) = - f(x)

• Even function:f(-x) = + f(x)

Graphical Example (odd)

Graphical Example (even)

Is sine odd or even?

• Graph the base graph of sine and determine if it is odd or even

Is sine odd or even?

Is sine odd or even?

• Sine is odd• On your green sheet of trig rules, find the odd-

even properties and write:sin(- x) = - sin(x)

Is cosine odd or even?

• Graph the base graph of cosine and determine if it is odd or even

Is cosine odd or even?

Is cosine odd or even?

• Cosine is even• On the green sheet write:

cos(- x) = cos(x)

Is tangent odd or even?

• Graph the base graph of tangent and determine if it is odd or even

Is tangent odd or even?

Is tangent odd or even?

• Tangent is odd• On your green sheet write:

tan(- x) = - tan(x)

What about the other three?

• The other three functions (secant, cosecant, and cotangent) will have the same property as its reciprocal

• On your green sheet add the red part:sin(- x) = - sin(x) (and csc)cos(- x) = cos(x) (and sec)tan(- x) = - tan(x) (and cot)

Using the odd-even properties

• Find the exact value of sin(-45°))45sin(

)45sin(

even)(odd

2

2

)y"" with goes sine circle,(unit

Using the odd-even properties

• Find the exact value of cos(-120°))120cos(

)120cos(

even)(odd

2

1

)x"" with goes cosine circle,(unit

Using the odd-even properties

• Find the exact value of)

3

2tan(

)3

2tan(

)3

2tan(

even)(odd

1

3

y/x)or slope"" with goes tangent circle,(unit

3

Using the odd-even properties(try this on your own)

• Find the exact value of

)3

2sec(

)3

2sec(

)3

2sec(

cosine) of reciprocal even,(odd

2

1of reciprocal

)x"" with goes cosine circle,(unit

2

Another Example

• If f(x) = cos(x) and f(a) = ¼, find the exact value of f(-a)

• Answer:• Since cosine is even, f(-a) = f(a)

Since f(a) = ¼, f(-a) = ¼

Example continued

• If f(x) = cos(x) and f(a) = ¼, find the exact value of f(a) + f(a + 2π)

• Answer:• Since the period of cosine is 2π, f(a + 2π) will

equal f(a) (think about the graph and how cos(0) = cos(2π) )

• So we have f(a) + f(a + 2π) = ¼ + ¼ = 2/4 = ½

HW Time

• You may use the rest of the period to work on homework