8.3 InveRSe TRIgOnOmeTRIC FUnCTIOnS Understanding and Using ...

Unit 3 Lesson 4: Trigonometric Identities and Equations

Sunday, September 17, 2023 11:59 PM

Learning Objectives

? Deduce a given trigonometric identity ? Simplify trigonometric expressions using trigonometric identities ?

Fundamental Trigonometric Identities

The quotient identities

sin()

cos(x)

tan() = ; cot() =

cos()

sin()

The reciprocal identities

1

1

1

sin() = csc() cos() = sec() ; tan() = cot()

1

1

1

csc() = ; sec() = ; cot() =

sin()

cos()

tan()

Pythagorean Identities

sin () + cos () = 1 tan () + 1 = sec () 1 + cot () = csc ()

The cofunction Identities

cos - = sin()

2 cot 2- = tan() csc - = sec() 2

Other fundamental identities

Lesson Study Guides Page 1

sin( + ) = sin cos + cos sin cos( + ) = cos cos - sin sin

sin( - ) = sin cos - cos sin cos( - ) = cos cos + sin sin

sin(-) = - sin() ; cos(-) = cos() ; tan(-) = -tan() csc(-) = - csc() ; sec(-) = sec() ; cot(-) = -cot()

Useful trigonometric identities that can be derived from the fundamental identities

sin 2 = 2 sin cos ; cos 2 = cos - sin

1

1

sin = (1 - cos 2) cos = (1 + cos 2)

2

2

tan + tan tan( + ) = 1-tan tan ;

tan - tan tan( - ) = 1+tan tan

sin 2- = cos cos 2- = sin

1 cos cos = [cos( + ) + cos( - )]

2

1 sin cos = [sin( + ) + sin( - )]

2

1 sin sin = [cos( - ) - cos( + )]

2

Law of Cosines: = + - 2 cos , where is the angle between &

Law of Sines: = = , where is side opposite angle , etc.

Derivations of the many of these identities are included in the video portion of the lesson. Example 3.4.1: Deduce tan + 1 = sec from sin + cos = 1

Lesson Study Guides Page 2

Example 3.4.2: Deduce tan( + ) = using fundamental identities

Using Identities to evaluate trigonometric functions

The identities can used to evaluate trigonometric expressions at angles other than the well known angles by rewriting the trigonometric expression using identities Example 3.4.3 Use an identity to evaluate sin(15?)

Lesson Study Guides Page 3

Simplifying trigonometric expressions using identities

Example 3.4.4 Simplify the trigonometric function using identities 2 sin() cos() + (sin() - cos())

sec()

Example 3.4.5 Simplify the trigonometric function using identities 1 + tan () + sec () cot ()

csc () + cot () csc ()

Lesson Study Guides Page 4

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