DoubleAngle and 11 5 HalfAngle 11 5 HalfAngle

Double-Angle and 11 -5 Half-Angle 11 -5 Half-Angle. Identities Warm Up Lesson Presentation Lesson Quiz Holt. Mc. Dougal Algebra 2 Holt

Double-Angle and 11 -5 Half-Angle Identities Warm Up Find tan θ for 0 ≤ θ ≤ 90°, if 1. 2. 3. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Objective Evaluate and simplify expressions by using double-angle and half-angle identities. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities You can use sum identities to derive the double-angle identities. sin 2θ = sin(θ + θ) = sinθ cosθ + cosθ sinθ = 2 sinθ cosθ Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities 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 sin 2θ + cos 2θ = 1. It is common to rewrite expressions as functions of θ only. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Example 1: Evaluating Expressions with Double-Angle Identities Find sin 2θ and tan 2θ if sinθ = and 0°<θ<90°. Step 1 Find cosθ to evaluate sin 2θ = 2 sinθcosθ. Method 1 Use the reference angle. In Ql, 0° < θ < 90°, and sinθ = x 2 + 2 2 = 5 2 Use the Pythagorean Theorem. Solve for x. r=5 θ x Holt Mc. Dougal Algebra 2 y=2

Double-Angle and 11 -5 Half-Angle Identities Example 1 Continued Method 2 Solve cos 2θ = 1 – sin 2θ cosθ = Substitute Simplify. Holt Mc. Dougal Algebra 2 for cosθ.

Double-Angle and 11 -5 Half-Angle Identities Example 1 Continued Step 2 Find sin 2θ = 2 sinθcosθ Apply the identity for sin 2θ. Substitute for cosθ. Simplify. Holt Mc. Dougal Algebra 2 for sinθ and

Double-Angle and 11 -5 Half-Angle Identities Example 1 Continued Step 3 Find tanθ to evaluate tan 2θ = . Apply the tangent ratio identity. Substitute for cosθ. Simplify. Holt Mc. Dougal Algebra 2 for sinθ and

Double-Angle and 11 -5 Half-Angle Identities Example 1 Continued Step 4 Find tan 2θ. Apply the identity for tan 2θ. Substitute Holt Mc. Dougal Algebra 2 for tan θ.

Double-Angle and 11 -5 Half-Angle Identities Example 1 Continued Step 4 Continued Simplify. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Caution! The signs of x and y depend on the quadrant for angle θ. sin cos Ql + + Qll + – Qlll – – Ql. V – + Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 1 Find tan 2θ and cos 2θ if cosθ = 270°<θ<360°. and Step 1 Find tanθ to evaluate tan 2θ = . Method 1 Use the reference angle. In Ql. V, 270° < θ < 360°, and cosθ = 12 + y 2 = 3 2 Use the Pythagorean Theorem. x=1 Solve for y. θ r=3 Holt Mc. Dougal Algebra 2 y= – 2√ 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 1 Continued Step 2 Find tan 2θ = Apply the identity for tan 2θ. Substitute – 2 Simplify. Holt Mc. Dougal Algebra 2 for tanθ.

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 1 Continued Step 3 Find cos 2θ = 2 cos 2θ – 1 Apply the identity for cos 2θ. Substitute Simplify. Holt Mc. Dougal Algebra 2 for cosθ.

Double-Angle and 11 -5 Half-Angle Identities You can use double-angle identities to prove trigonometric identities. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Example 2 A: Proving identities with Double-Angle Identities Prove each identity. sin 2θ = 2 tanθ – 2 tanθ sin 2θ Choose the right-hand side to modify. = 2 tanθ (1– sin 2θ) = 2 tanθ cos 2θ = 2(tanθcosθ)cosθ = 2 sinθcosθ = sin 2θ Holt Mc. Dougal Algebra 2 Factor 2 tanθ. Rewrite using 1 –sin 2θ = cos 2θ. Regroup. Rewrite using tanθcosθ = sinθ. Apply the identity for sin 2θ.

Double-Angle and 11 -5 Half-Angle Identities Example 2 B: Proving identities with Double-Angle Identities cos 2θ = (2 – sec 2θ)(1 – sin 2θ) = (2 – sec 2θ)(cos 2θ) Choose the right-hand side to modify. Rewrite using 1 – sin 2θ = cos 2θ. = 2 cos 2θ – 1 Expand simplify. = cos 2θ Apply the identity for cos 2θ. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Helpful Hint Choose to modify either the left side or the right side of an identity. Do not work on both sides at once. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 2 a Prove each identity. cos 4θ – sin 4θ = cos 2θ (cos 2θ – sin 2θ)(cos 2θ + sin 2θ) = (1)(cos 2θ) = Factor the left side. Rewrite using 1 = cos 2θ + sin 2θ and cos 2θ = cos 2θ – sin 2θ. cos 2θ = cos 2θ Simplify. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 2 b Prove each identity. Rewrite tan θ ratio identity and Pythagorean identity. Reciprocal sec θ identity and simplify fraction. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 2 b Continued Prove each identity. Simplify. Double angle identity. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities You can use double-angle identities for cosine to derive the half-angle identities by substituting for θ. For example, cos 2θ = 2 cos 2θ – 1 can be rewritten as cosθ = 2 cos 2 – 1. Then solve for cos Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Half-angle identities are useful in calculating exact values for trigonometric expressions. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Example 3 A: Evaluating Expressions with Half-Angle Identities Use half-angle identities to find the exact value of cos 15°. Positive in Ql. Cos 30° = Simplify. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Example 3 A Continued Check Use your calculator. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Example 3 B: Evaluating Expressions with Half-Angle Identities Use half-angle identities to find the exact value of . Negative in Qll. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Example 3 B Continued Simplify. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Example 3 B Continued Check Use your calculator. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 3 a Use half-angle identities to find the exact value of tan 75°. tan (150°) Positive in Ql. Simplify. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 3 a Continued Check Use your calculator. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 3 b Use half-angle identities to find the exact value of . Negative in Qll. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 3 b Continued Simplify. Check Use your calculator. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities 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 < 242 + 72 = x 2 Thus, cosθ = Holt Mc. Dougal Algebra 2 and tanθ = Pythagorean Theorem. Solve for the missing side x.

Double-Angle and 11 -5 Half-Angle Identities Example 4 Continued Step 2 Evaluate cos x θ Choose + for cos where 0 < θ < Evaluate. Holt Mc. Dougal Algebra 2 24 7

Double-Angle and 11 -5 Half-Angle Identities Example 4 Continued Simplify. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Example 4 Continued Step 3 Evaluate tan Choose + for tan 0<θ< Evaluate. Holt Mc. Dougal Algebra 2 where

Double-Angle and 11 -5 Half-Angle Identities Example 4 Continued Simplify. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 4 Find sin and cos if tan θ = and 0 < θ < 90. Step 1 Find cos θ to evaluate the half-angle identities. Use the reference angle. In Ql, 0 < θ < 42 + 3 2 = r and tanθ = Pythagorean Theorem. 2 r= Thus, cosθ = Holt Mc. Dougal Algebra 2 . Solve for the missing side r.

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 4 Continued Step 2 Evaluate cos r θ Choose + for cos where 0 < θ < Evaluate. Simplify. Holt Mc. Dougal Algebra 2 3 4

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 4 Continued Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 4 Continued Step 3 Evaluate sin Choose + for sin 90°. Evaluate. Holt Mc. Dougal Algebra 2 where 0 < θ <

Double-Angle and 11 -5 Half-Angle Identities Check It Out! Example 4 Continued Simplify. Holt Mc. Dougal Algebra 2

Double-Angle and 11 -5 Half-Angle Identities Lesson Quiz: Part I 1. Find cos and cos 2θ if sin θ = 2. Prove the following identity: Holt Mc. Dougal Algebra 2 and 0 < θ <

Double-Angle and 11 -5 Half-Angle Identities Lesson Quiz: Part II 3. Find the exact value of cos 22. 5°. Holt Mc. Dougal Algebra 2