2312 - Section 4.4 - Trigonometric Expressions and Identities - UH

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2312 - Section 4.4 - Trigonometric Expressions and Identities

In this section we are going to practice the algebra involved in working with the trigonometric functions.

Notational Conventions

1. An expression such as sin really means sin().

An exception to this, however, occurs in expressions such as sin( + ),

where the parentheses are necessary.

Example: sin ( + ) sin +

42

42

2. Parentheses are often omitted in multiplication. For example: (sin )(cos ) is usually written sin cos .

3. The quantity (sin ) = sin = sin sin ... sin .

Example: (sin )2 = sin2 = sin sin . sin2 sin 2

In simplifying expressions, it may be useful to use the following identities.

Basic Trigonometric Identities

Reciprocal Identities 1. sec = 1 , cos 0

cos

csc = 1 , sin 0

sin

cot = 1 , tan 0

tan

2. sin = tan ; cos = cot

cos

sin

Pythagorean Identities

3. sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2

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Example: Simplify. (sin - cos )2 + 2 sin cos

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Recall: ( + )2 = 2 + 2 + 2 ( - )2 = 2 - 2 + 2

Example: Simplify.

(1 - cos )(csc + cot )

Example: Simplify.

sin4 - 2 sin2 + 1

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Example: Simplify.

sin (cot + tan )

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Example: Simplify.

cos2 + sin2 + cot2

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Example: Simplify.

3 cos2 + sec2 + 3 sin2 - tan2

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Example: Simplify.

tan sec +1

tan

tan

+

sec + 1 sec - 1

+ tan

sec -1

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