Vocabulary identity trigonometric identity cofunction odd-even identities BELLRINGER: Define each word in your notebook.

Key Concept 1

Example 1 Use Reciprocal and Quotient Identities A. If, find sec θ.Divide. Reciprocal Identity Answer:

Example 1 Use Reciprocal and Quotient Identities B. If and, find sin x. Reciprocal Identity Quotient Identity Substitute for cos x.

Example 1 Use Reciprocal and Quotient Identities Answer: Divide. Multiply each side by. Simplify.

Key Concept 2

Example 2 Use Pythagorean Identities If cot θ = 2 and cos θ < 0, find sin θ and cos θ. cot 2 θ + 1= csc 2 θ Pythagorean Identity (2) 2 + 1= csc 2 θcot θ = 2 5= csc 2 θ Simplify. Use the Pythagorean Identity that involves cot θ. = csc θ Take the square root of each side. Reciprocal Identity Solve for sin θ.

Example 2 Use Pythagorean Identities Since is positive and cos θ < 0, sin θ must be negative. So. You can then use this quotient identity again to find cos θ. Quotient Identitycot θ = 2 and Multiply each side by.

Example 2 Use Pythagorean Identities So, Check sin 2 θ + cos 2 θ= 1Pythagorean Identity Answer: Simplify.

Key Concept 3

Key Concept 4

Example 3 Use Cofunction and Odd-Even Identities Simplify. cos x = –0.75 Cofunction Identity If cos x = –0.75, find Odd-Even Identity Factor.

Example 3 Use Cofunction and Odd-Even Identities Answer: 0.75 So, = 0.75.

Example 4 Simplify by Rewriting Using Only Sine and Cosine Solve Algebraically Simplify. Multiply. Pythagorean Identity So, = cos x.

Example 5 Simplify by Factoring Simplify cos x tan x – sin x cos 2 x. Solve Algebraically cos x tan x – sin x cos 2 xOriginal expression Factor. Multiply. Quotient Identity So, cos x tan x – sin x cos 2 x = sin 3 x. Pythagorean Identity Simplify.= sin 3 x

Example 6 Simplify by Combining Fractions Common denominator Multiply. Add the numerators. Simplify. Pythagorean Identity Simplify.

Example 6 Answer: – 2 sec 2 x – 2 sec 2 x Simplify by Combining Fractions Reciprocal Identity –2csc 2 x. Divide out common factor. Reciprocal and Quotient Identities

Example 7 Rewrite to Eliminate Fractions Rewrite as an expression that does not involve a fraction. Pythagorean Identity Reciprocal Identity Quotient Identity Reciprocal Identity

Example 7 Rewrite to Eliminate Fractions Answer: tan 2 x So, = tan 2 x.