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Name_____________________________________ Class____________________________ Date________________ Lesson 14-7 Double-Angle and Half-Angle Identities Lesson Objectives 1 Verifying and using double angle identities 2 Verifying and using half-angle identities NAEP 2005 Strand: Algebra Topic: Variables, Expressions, and Operations Local Standards: ____________________________________ Key Concepts. 2 cos 2θ 2 cos 2θ 2 cos 2θ 2 θ θ 2 2 θ θ tan 2θ θ θ sin 2θ 2 2 θ 1 2 θ All rights reserved. Double-Angle Identities 1 sin A 2 A 2 cos A 2 1 tan A 2 1 A 1 A A 2 Examples. 1 Using a Double-Angle Identity Use a double-angle identity to find the exact value of sin 600°. sin 600° sin (2 300°) 2 ( 2 292 Rewrite 600 as (2 300). 300° )( Algebra 2 Lesson 14-7 300° ) Use a sine -angle identity. Replace with exact values. Simplify. Daily Notetaking Guide © Pearson Education, Inc., publishing as Pearson Prentice Hall. Half-Angle Identities Name_____________________________________ Class____________________________ Date ________________ 2 Verifying an Identity Verify the identity sin 2θ (1 sin2 θ) θ 2 θ -angle and θ identities θ 2 Simplify. θ 2 All rights reserved. sin 2u 2 tan θ. (1 2 sin 2 u) Identity θ 3 Using Half-Angle Identities Use the half-angle identities to find the exact value of sin 75˚. sin 75 sin 150 Rewrite 75 as 2 ( 150 2 1 ( 2 4 ) ) ( ). 150 2 Use the since sin 75 is positive. Substitute the exact value for square root, 150. Simplify. © Pearson Education, Inc., publishing as Pearson Prentice Hall. Simplify. Quick Check. 1. Use a double-angle identity to find the exact value of sin 120°. 2. Verify the identity 2 cos 2θ 4 cos2 θ 2. Daily Notetaking Guide Algebra 2 Lesson 14-7 293 Name_____________________________________ Class____________________________ Date ________________ Example. 12 and 90° θ 180°, find sin u. 4 Using Half-Angle Identities Given cos θ 13 2 Since 90° θ 180°, 2u θ ( 2 -angle identity 2 1 25 26 26 26 . ) Substitute. Choose the square root since θ 2 is in Quadrant . Simplify. All rights reserved. sin θ 2 and 2u is in Quadrant Rationalize the denominator. Quick Check. 4. Use the information from Example 4 to find exact values of cos 2u and tan 2u. 294 Algebra 2 Lesson 14-7 Daily Notetaking Guide © Pearson Education, Inc., publishing as Pearson Prentice Hall. 3. Use the half-angle identities to find the exact value of each expression. a. sin 150° b. tan 150°