Precalculus Fifth Edition Mathematics for Calculus James Stewart Lothar Redlin Saleem Watson.

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Presentation transcript:

Precalculus Fifth Edition Mathematics for Calculus James Stewart Lothar Redlin Saleem Watson

Trigonometric Identities 7.1

Fundamental Trigonometric Identities Reciprocal Identities

Fundamental Trigonometric Identities Pythagorean Identities sin 2 x + cos 2 x = 1 tan 2 x + 1 = sec 2 x 1 + cot 2 x = csc 2 x

Fundamental Trigonometric Identities Even-Odd Identities sin( – x) = – sin x cos( – x) = cos x tan( – x) = – tan x

Simplifying Trigonometric Expressions

Identities enable us to write the same expression in different ways. It is often possible to rewrite a complicated looking expression as a much simpler one.

Simplifying Trigonometric Expressions To simplify algebraic expressions, we used: Factoring Common denominators Special Product Formulas

Simplifying Trigonometric Expressions To simplify trigonometric expressions, we use these same techniques together with the fundamental trigonometric identities.

E.g. 1—Simplifying a Trigonometric Expression Simplify the expression cos t + tan t sin t We start by rewriting the expression in terms of sine and cosine.

E.g. 1—Simplifying a Trigonometric Expression

E.g. 2—Simplifying by Combining Fractions Simplify the expression We combine the fractions by using a common denominator.

E.g. 2—Simplifying by Combining Fractions

More Examples: Simplify:

More Examples: Simplify:

Derivatives of trigonometric functions Find and simplify: