8.3 InveRSe TRIgOnOmeTRIC FUnCTIOnS Understanding and ...

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(?)

tan(?) = ?????? ;

cos(?)

cos(x)

cot(?) = ??????

sin(?)

The reciprocal identities

1

sin(?) = ?????? ¡Ã

csc(?)

1

cos(?) = ?????? ;

sec(?)

1

tan(?) = ??????

cot(?)

1

csc(?) = ?????? ;

sin(?)

1

sec(?) = ?????? ;

cos(?)

1

cot(?) = ??????

tan(?)

Pythagorean Identities

sin (?) + cos (?) = 1

tan (?) + 1 = sec (?)

1 + cot (?) = csc (?)

The cofunction Identities

?

cos ??? ? = sin(?)

2

?

cot ??? ? = tan(¦È)

2

?

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(?) ;

csc(??) = ? csc(?) ; sec(??) = sec(?) ;

tan(??) = ?tan(?)

cot(??) = ?cot(?)

Useful trigonometric identities that can be derived from the fundamental identities

sin 2? = 2 sin ? cos ?

;

1

sin ? = ??(1 ? cos 2?)

2

cos 2? = cos ? ? sin ?

¡Ã

tan ? + tan ?

tan(? + ?) = ?????????????

1 ? tan ? tan ?

?

sin ??? ? = cos ?

2

¡Ã

1

cos ? = ??(1 + cos 2?)

2

;

tan ? ? tan ?

tan(? ? ?) = ?????????????

1 + tan ? tan ?

?

cos ??? ? = sin ?

2

1

cos ? cos ? = ??[cos(? + ?) + cos(? ? ?)]

2

¡Ã

1

sin ? sin ? = ??[cos(? ? ?) ? cos(? + ?)]

2

1

sin ? cos ? = ??[sin(? + ?) + sin(? ? ?)]

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¡ã)

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

Lesson Study Guides Page 5

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