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# Chapter 5: Analytic Trigonometry - PowerPoint PPT Presentation

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

Section 5.1a: Fundamental Identities

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

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.

Reciprocal Identities

Quotient Identities

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

identities. Which three?

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?

Reciprocal Identities

Quotient Identities

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

equation are undefined at . Which three?

Recall our unit circle:

P

What are the coordinates of P?

sint

(1,0)

cost

So by the Pythagorean Theorem:

Divide by :

Recall our unit circle:

P

What are the coordinates of P?

sint

(1,0)

cost

So by the Pythagorean Theorem:

Divide by :

Given and , find and .

We only take the positive answer…why?

Cofunction Identities

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

If , find .

Sine is odd 

Cofunction Identity 

Simplify the given expression.

How can we support this answer graphically???

Simplify the given expression.

Graphical support?

Simplify the given expressions to either a constant or a basic

trigonometric function. Support your result graphically.

Simplify the given expressions to either a constant or a basic

trigonometric function. Support your result graphically.

Use the basic identities to change the given expressions to ones

involving only sines and cosines. Then simplify to a basic

trigonometric function.

Use the basic identities to change the given expressions to ones

involving only sines and cosines. Then simplify to a basic

trigonometric function.

Use the basic identities to change the given expressions to ones

involving only sines and cosines. Then simplify to a basic

trigonometric function.

Simplify the expression

graphical support?

basic trigonometric function.

basic trigonometric function.

basic trigonometric function.

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:

expression of a single trigonometric function.

e.g.,

Let

Substitute:

Factor:

expression of a single trigonometric function.

e.g.,

expression of a single trigonometric function.

e.g.,

Let

expression of a single trigonometric function.

e.g.,

trigonometric function.

e.g.,