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Chapter 5: Analytic Trigonometry

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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

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.

Basic Trigonometric Identities

Reciprocal Identities

Quotient Identities

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

identities. Which three?

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?

Basic Trigonometric Identities

Reciprocal Identities

Quotient Identities

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

equation are undefined at . Which three?

Pythagorean Identities

Recall our unit circle:

P

What are the coordinates of P?

sint

(1,0)

cost

So by the Pythagorean Theorem:

Divide by :

Pythagorean Identities

Recall our unit circle:

P

What are the coordinates of P?

sint

(1,0)

cost

So by the Pythagorean Theorem:

Divide by :

Pythagorean Identities

Given and , find and .

We only take the positive answer…why?

Cofunction Identities

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

Odd-Even Identities

If , find .

Sine is odd

Cofunction Identity

Simplifying Trigonometric Expressions

Simplify the given expression.

How can we support this answer graphically???

Simplifying Trigonometric Expressions

Simplify the given expression.

Graphical support?

Simplifying Trigonometric Expressions

Simplify the given expressions to either a constant or a basic

trigonometric function. Support your result graphically.

Simplifying Trigonometric Expressions

Simplify the given expressions to either a constant or a basic

trigonometric function. Support your result graphically.

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.

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.

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.

Let’s start with a practice problem…

Simplify the expression

How about some

graphical support?

Combine the fractions and simplify to a multiple of a power of a

basic trigonometric function.

Combine the fractions and simplify to a multiple of a power of a

basic trigonometric function.

Combine the fractions and simplify to a multiple of a power of a

basic trigonometric function.

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:

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:

Write each expression in factored form as an algebraic

expression of a single trigonometric function.

e.g.,

Write each expression in factored form as an algebraic

expression of a single trigonometric function.

e.g.,

Let

Write each expression in factored form as an algebraic

expression of a single trigonometric function.

e.g.,

Write each expression as an algebraic expression of a single

trigonometric function.

e.g.,