Holt Algebra 2

UNIT 10.1 UNIT 10.1 TRIGONOMETRICTRIGONOMETRIC

IDENTITIESIDENTITIES

Warm Up

Simplify.

1.

2.

cos A

1

Use fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities.

Objective

You can use trigonometric identities to simplify trigonometric expressions. Recall that an identity is a mathematical statement that is true for all values of the variables for which the statement is defined.

A derivation for a Pythagorean identity is shown below.

x2 + y2 = r2

cos2 θ + sin2 θ = 1

Pythagorean Theorem

Divide both sides by r2.

Substitute cos θ for and

sin θ for

To prove that an equation is an identity, alter one side of the equation until it is the same as the other side. Justify your steps by using the fundamental identities.

Example 1A: Proving Trigonometric Identities

Prove each trigonometric identity.

Choose the right-hand side to modify.

Reciprocal identities.

Simplify.

Ratio identity.

Example 1B: Proving Trigonometric Identities

Prove each trigonometric identity.

1 – cot θ = 1 + cot(–θ)

= 1 + (–cotθ)

= 1 – cotθ

Choose the right-hand side to modify.

Reciprocal identity.

Negative-angle identity.

Reciprocal identity.

Simplify.

You may start with either side of the given equation. It is often easier to begin with the more complicated side and simplify it to match the simpler side.

Helpful Hint

Check It Out! Example 1a

Prove each trigonometric identity.

sin θ cot θ = cos θ

cos θ = cos θ

Choose the left-hand side to modify.

Ratio identity.

Simplify.

cos θ

Check It Out! Example 1b

Prove each trigonometric identity.

1 – sec(–θ) = 1 – secθ Choose the left-hand side to modify.

Reciprocal identity.

Negative-angle identity.

Reciprocal Identity.

You can use the fundamental trigonometric identities to simplify expressions.

If you get stuck, try converting all of the trigonometric functions to sine and cosine functions.

Helpful Hint

Example 2A: Using Trigonometric Identities to Rewrite Trigonometric Expressions

Rewrite each expression in terms of cos θ, and simplify.

sec θ (1 – sin2θ)

cos θ

Substitute.

Multiply.

Simplify.

Example 2B: Using Trigonometric Identities to Rewrite Trigonometric Expressions

Rewrite each expression in terms of sin θ, cos θ, and simplify.

sinθ cosθ(tanθ + cotθ)

sin2θ + cos2θ

Substitute.

Multiply.

Simplify.

1 Pythagorean identity.

Check It Out! Example 2a

Rewrite each expression in terms of sin θ, and simplify.

Pythagorean identity.

Simplify.

Factor the difference of two squares.

Check It Out! Example 2b

Rewrite each expression in terms of sin θ, and simplify.

cot2θ

csc2θ – 1 Pythagorean identity.

Substitute.

Simplify.

Example 3: Physics Application

At what angle will a wooden block on a concrete incline start to move if the coefficient of friction is 0.62?

Set the expression for the weight component equal to the expression for the force of friction.

mg sinθ = μmg cosθ

sinθ = μcosθ

sinθ = 0.62 cosθ

Divide both sides by mg.

Substitute 0.62 for μ.

Example 3 Continued

tanθ = 0.62

θ = 32°

The wooden block will start to move when the concrete incline is raised to an angle of about 32°.

Divide both sides by cos θ.

Ratio identity.

Evaluate inverse tangent.

Check It Out! Example 3

Use the equation mg sinθ = μmg cosθ to determine the angle at which a waxed wood block on a wood incline with μ = 0.4 begins to slide.

Set the expression for the weight component equal to the expression for the force of friction.

mg sinθ = μmg cosθ

sinθ = μcosθ

sinθ = 0.4 cosθ

Divide both sides by mg.

Substitute 0.4 for μ.

Check It Out! Example 3 Continued

tanθ = 0.4

θ = 22°

The wooden block will start to move when the concrete incline is raised to an angle of about 22°.

Divide both sides by cos θ.

Ratio identity.

Evaluate inverse tangent.

Lesson Quiz: Part I

1. sinθ secθ =

Prove each trigonometric identity.

2. sec2θ = 1 + sin2θ sec2θ

= 1 + tan2θ

= sec2θ

Lesson Quiz: Part II

Rewrite each expression in terms of cos θ, and simplify.

3. sin2θ cot2θ secθ cosθ

4. 2 cosθ

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