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Chapter 5: Analytic Trigonometry. Section 5.1a: Fundamental Identities HW: p. 451-452 1-7 odd, 27-49 odd. Is this statement true?. This identity is a true sentence, but only w ith the qualification that x must be in the d omain of both expressions .

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chapter 5 analytic trigonometry

Chapter 5: Analytic Trigonometry

Section 5.1a: Fundamental Identities

HW: p. 451-452 1-7 odd, 27-49 odd

slide2
Is this statement true?

This identity is a true sentence, but only

with the qualification that x must be in the

domain of both expressions.

If either side of the equality is undefined (i.e., at x = –1), then

the entire expression is meaningless!!!

The statement is a trigonometric identity

because it is true for all values of the variable for which both

sides of the equation are defined.

The set of all such values is called the domain of validity of

the identity.

slide3
Basic Trigonometric Identities

Reciprocal Identities

Quotient Identities

is in the domain of validity of exactly three of the basic

identities. Which three?

slide4
Basic Trigonometric Identities

Reciprocal Identities

Quotient Identities

For exactly two of the basic identities, one side of the equation

is defined at and the other side is not. Which two?

slide5
Basic Trigonometric Identities

Reciprocal Identities

Quotient Identities

For exactly three of the basic identities, both sides of the

equation are undefined at . Which three?

slide6
Pythagorean Identities

Recall our unit circle:

P

What are the coordinates of P?

sint

(1,0)

cost

So by the Pythagorean Theorem:

Divide by :

slide7
Pythagorean Identities

Recall our unit circle:

P

What are the coordinates of P?

sint

(1,0)

cost

So by the Pythagorean Theorem:

Divide by :

slide8
Pythagorean Identities

Given and , find and .

We only take the positive answer…why?

slide9
Cofunction Identities

Can you explain why each of these is true???

slide10
Odd-Even Identities

If , find .

Sine is odd 

Cofunction Identity 

slide11
Simplifying Trigonometric Expressions

Simplify the given expression.

How can we support this answer graphically???

slide12
Simplifying Trigonometric Expressions

Simplify the given expression.

Graphical support?

slide13
Simplifying Trigonometric Expressions

Simplify the given expressions to either a constant or a basic

trigonometric function. Support your result graphically.

slide14
Simplifying Trigonometric Expressions

Simplify the given expressions to either a constant or a basic

trigonometric function. Support your result graphically.

slide15
Simplifying Trigonometric Expressions

Use the basic identities to change the given expressions to ones

involving only sines and cosines. Then simplify to a basic

trigonometric function.

slide16
Simplifying Trigonometric Expressions

Use the basic identities to change the given expressions to ones

involving only sines and cosines. Then simplify to a basic

trigonometric function.

slide17
Simplifying Trigonometric Expressions

Use the basic identities to change the given expressions to ones

involving only sines and cosines. Then simplify to a basic

trigonometric function.

slide18
Let’s start with a practice problem…

Simplify the expression

How about some

graphical support?

slide22
Quick check of your algebra skills!!!

Factor the following expression (without any guessing!!!)

What two numbers have a product of –180 and a sum of 8?

Rewrite middle term:

Group terms and factor:

Divide out common factor:

slide23
Write each expression in factored form as an algebraic

expression of a single trigonometric function.

e.g.,

Let

Substitute:

Factor:

“Re”substitute for your answer:

slide24
Write each expression in factored form as an algebraic

expression of a single trigonometric function.

e.g.,

slide25
Write each expression in factored form as an algebraic

expression of a single trigonometric function.

e.g.,

Let

slide26
Write each expression in factored form as an algebraic

expression of a single trigonometric function.

e.g.,

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