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Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

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Page 1: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:
Page 2: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

Section 5.1

Simplifying More Trig Expressions

Page 3: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

Fundamental Trigonometric Identities:Reciprocal Identities:

1sin

csc

1cos

sec

1tan

cot

1csc

sin

1

seccos

1

cottan

Quotient Identities:

sintan

cos

coscot

sin

Pythagorean Identities:2 2sin cos 1 2 21 tan sec 2 21 cot csc

Cofunction Identities:

sin cos2

cos sin2

tan cot2

cot tan2

sec csc2

csc sec2

Page 4: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

Even / Odd Identities:

sin( ) sin

csc( ) csc

sec( ) sec

cos( ) cos

tan( ) tan

cot( ) cot

Page 5: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

Trig Identities

•Any relationship that is true for all values of the variable for which each side is defined is called an identity. 

•We can use trig identities to simplify trig expressions, prove other identities and solve more complex trig equations.

Page 6: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

Strategies for Simplifying Trig Expressions:

1.

2.

3.

4.

5.

Page 7: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

We can factor trigonometric expressions, too!

Ex 1: Factor each trig expression

A. B. C. 2sec 1 24 tan tan 3 2csc 2csc 3

Page 8: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

sin cotx x

Ex 2: Use trig identities to simplify

Page 9: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

2 2csc 1 cosx x

Ex 3: Use trig identities to simplify

Page 10: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

Ex 4: Simplify by factoring:

2cos 4

cos 2

x

x

Page 11: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

Ex 5: Use trig identities to simplify

2sin cos sinx x x

Page 12: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

Ex 6: Simplify by adding the fractions first:

1 1

sec 1 sec 1x x

Page 13: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

Ex 7: Simplify by factoring

4 2csc 2csc 1

Page 14: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

Ex 8: Simplify by adding the fractions first:

sec tan

cos cot

x x

x x

Page 15: Section 5.1 Simplifying More Trig Expressions Fundamental Trigonometric Identities: Reciprocal Identities: Quotient Identities: Pythagorean Identities:

HomeworkPage 379

(2, 27-36 multiples of 3, 45-53 odd, 61, 64, 65)