Algebra 2 unit 10.6

46
UNIT 10.7 DOUBLE-ANGLE AND UNIT 10.7 DOUBLE-ANGLE AND HALF-ANGLE IDENTITIES HALF-ANGLE IDENTITIES

Transcript of Algebra 2 unit 10.6

UNIT 10.7 DOUBLE-ANGLE ANDUNIT 10.7 DOUBLE-ANGLE ANDHALF-ANGLE IDENTITIESHALF-ANGLE IDENTITIES

Warm Up

Find tan θ for 0 ≤ θ ≤ 90°, if

1.

2.

3.

Evaluate and simplify expressions by using double-angle and half-angle identities.

Objective

You can use sum identities to derive the double-angle identities.

sin 2θ = sin(θ + θ)

= sinθ cosθ + cosθ sinθ

= 2 sinθ cosθ

You can derive the double-angle identities for cosine and tangent in the same way. There are three forms of the identity for cos 2θ, which are derived by using sin2θ + cos2θ = 1. It is common to rewrite expressions as functions of θ only.

Example 1: Evaluating Expressions with Double-Angle Identities

Find sin2θ and tan2θ if sinθ = and 0°<θ<90°.

Step 1 Find cosθ to evaluate sin2θ = 2sinθcosθ.

Method 1 Use the reference angle.

In Ql, 0° < θ < 90°, and sinθ =

x2 + 22 = 52

θ

r = 5y = 2

x

Use the PythagoreanTheorem.

Solve for x.

Example 1 Continued

Method 2 Solve cos2θ = 1 – sin2θ.

cos2θ = 1 – sin2θ

cosθ =Substitute for cosθ.

Simplify.

Example 1 Continued

Step 2 Find sin2θ.

sin2θ = 2sinθcosθ Apply the identity for sin2θ.

Simplify.

Substitute for sinθ and

for cosθ.

Example 1 Continued

Step 3 Find tanθ to evaluate tan2θ = .

Apply the tangent ratio identity.

Simplify.

Substitute for sinθ and

for cosθ.

Example 1 Continued

Step 4 Find tan 2θ.

Apply the identity for tan2θ.

Substitute for tan θ.

Example 1 Continued

Step 4 Continued

Simplify.

The signs of x and y depend on the quadrant for angle θ.

sin cos

Ql + +

Qll + –

Qlll – –

QlV – +

Caution!

Find tan2θ and cos2θ if cosθ = and 270°<θ<360°.

Method 1 Use the reference angle.

Check It Out! Example 1

Step 1 Find tanθ to evaluate tan2θ = .

In QlV, 270° < θ < 360°, and cosθ =

12 + y2 = 32 Use the Pythagorean Theorem.

Solve for y. θ

r=3

x=1

y= –2√ 2

Check It Out! Example 1 Continued

Step 2 Find tan2θ.

Apply the identity for tan2θ.

Simplify.

tan2θ =

Substitute –2 for tanθ.

Check It Out! Example 1 Continued

Step 3 Find cos2θ.

cos2θ = 2cos2θ – 1 Apply the identity for cos2θ.

Simplify.

Substitute for cosθ.

You can use double-angle identities to prove trigonometric identities.

Example 2A: Proving identities with Double-Angle Identities

Prove each identity.

sin 2θ = 2tanθ – 2tanθ sin2θ Choose the right-hand side to modify.

= 2tanθ (1– sin2θ) Factor 2tanθ.

= 2tanθ cos2θRewrite using 1 –sin2θ = cos2θ.

= 2(tanθcosθ)cosθ Regroup.

= 2sinθcosθRewrite using tanθcosθ

= sinθ.

= sin2θ Apply the identity for sin2θ.

Example 2B: Proving identities with Double-Angle Identities

cos2θ = (2 – sec2θ)(1 – sin2θ)

cos2θ = (2 – sec2θ)(1 – sin2θ)

= (2 – sec2θ)(cos2θ)

= 2cos2θ – 1

= cos2θ

Choose the right-hand side to modify.

Rewrite using 1 – sin2θ = cos2θ.

Expand and simplify.

Apply the identity for cos2θ.

Choose to modify either the left side or the right side of an identity. Do not work on both sides at once.

Helpful Hint

Check It Out! Example 2a

cos4θ – sin4θ = cos2θ

(cos2θ – sin2θ)(cos2θ + sin2θ) =

(1)(cos2θ) =

cos2θ = cos2θ

Factor the left side.

Rewrite using 1 = cos2θ + sin2θ and cos2θ = cos2θ – sin2θ.

Simplify.

Prove each identity.

Check It Out! Example 2b Prove each identity.

Rewrite tan θ ratio identity and Pythagorean identity.

Reciprocal sec θ identity and simplify fraction.

Check It Out! Example 2b Continued

Prove each identity.

Simplify.

Double angle identity.

You can use double-angle identities for cosine to derive

the half-angle identities by substituting for θ. For

example, cos2θ = 2 cos2θ – 1 can be rewritten as cosθ = 2

cos2 – 1. Then solve for cos

Half-angle identities are useful in calculating exact values for trigonometric expressions.

Example 3A: Evaluating Expressions with Half-Angle Identities

Use half-angle identities to find the exact value of cos 15°.

Positive in Ql.

Simplify.

Cos 30° =

Example 3A Continued

Check Use your calculator.

Example 3B: Evaluating Expressions with Half-Angle Identities

Use half-angle identities to find the exact value

of .

Negative in Qll.

Example 3B Continued

Simplify.

Example 3B Continued

Check Use your calculator.

Check It Out! Example 3a Use half-angle identities to find the exact value of tan 75°.

tan (150°)

Positive in Ql.

Simplify.

Check It Out! Example 3a Continued

Check Use your calculator.

Check It Out! Example 3b

Use half-angle identities to find the exact value

of .

Negative in Qll.

Check It Out! Example 3b Continued

Simplify.

Check Use your calculator.

Example 4: Using the Pythagorean Theorem with Half-Angle Identities

Find cos and tan if tan θ = and 0<θ<

Step 1 Find cos θ to evaluate the half-angle identities. Use the reference angle.

In Ql, 0 < θ < and tanθ =

242 + 72 = x2

Thus, cosθ =

Pythagorean Theorem.

Solve for the missing side x.

Example 4 Continued

x7

24θ

Step 2 Evaluate cos

Evaluate.

Choose + for cos

where 0 < θ <

Simplify.

Example 4 Continued

Example 4 Continued

Step 3 Evaluate tan

Choose + for tan where

0 < θ <

Evaluate.

Example 4 Continued

Simplify.

Check It Out! Example 4

Step 1 Find cos θ to evaluate the half-angle identities. Use the reference angle.

42 + 32 = r 2 Pythagorean Theorem.

Solve for the missing side r.

Find sin and cos if tan θ = and 0 < θ < 90.

In Ql, 0 < θ < and tanθ =

r =

Thus, cosθ = .

Check It Out! Example 4 Continued

r4

Step 2 Evaluate cos

Evaluate.

Choose + for cos

where 0 < θ <

Simplify.

Check It Out! Example 4 Continued

Check It Out! Example 4 Continued

Step 3 Evaluate sin

Choose + for sin where 0 < θ

< 90°.

Evaluate.

Check It Out! Example 4 Continued

Simplify.

Lesson Quiz: Part I

1. Find cos and cos 2θ if sin θ = and 0 < θ <

2. Prove the following identity:

Lesson Quiz: Part II

3. Find the exact value of cos 22.5°.

All rights belong to their respective owners.Copyright Disclaimer Under Section 107 of the Copyright Act 1976, allowance is made for "fair use" for purposes such as criticism, comment, news reporting, TEACHING, scholarship, and research. Fair use is a use permitted by copyright statute that might otherwise be infringing. Non-profit, EDUCATIONAL or personal use tips the balance in favor of fair use.