1. Splash Screen

2. Lesson Menu Five-Minute Check (over Lesson 13–2) CCSS Then/Now Key Concept: Sum Identities/Difference Identities Example 1:Find Trigonometric Values Example 2:Real-World Example: Sum and Difference of Angles Identities Example 3:Verify Trigonometric Identities

3. Over Lesson 13–2 5-Minute Check 1 Find the missing step in the identity. ? A. B. C. D.

4. Over Lesson 13–2 5-Minute Check 2 Find the missing step in the identity. ? A. B. C. D.

5. Over Lesson 13–2 5-Minute Check 3 A.sin 2  B.cos 2  C.sin  D.cos  Simplify (1 – sin  )(1 + sin  ) using a trigonometric identity.

6. Over Lesson 13–2 5-Minute Check 4 A.sin 2  B.tan 2  C.cos 2  D.1

7. CCSS Content Standards F.TF.8 Prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to find sin (θ), cos (θ), or tan (θ) given sin (θ), cos (θ), or tan (θ) and the quadrant of the angle. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. 6 Attend to precision.

8. Then/Now You found values of trigonometric functions for general angles. Find values of sine and cosine by using sum and difference identities. Verify trigonometric identities by using sum and difference identities.

9. Concept

10. Example 1 Find Trigonometric Values A. Find the exact value of sin 75°. sin 75° = sin (30° + 45°) = sin 30° cos 45° + cos 30° sin 45°Sum of angles Evaluate each expression.

11. Example 1 Find Trigonometric Values Multiply. Simplify.

12. Example 1 Find Trigonometric Values Evaluate each expression. B. Find the exact value of cos (–75°). cos (–75°) = cos (60° – 135°) = cos 60° cos 135° + sin 60° sin 135° Difference of angles

13. Example 1 Find Trigonometric Values Multiply. Simplify.

14. Example 1 A. Find the exact value of sin 105°. A. B. C. D.

15. Example 1 B. Find the exact value of cos (–120°). A. B. C. D.

16. Example 2 Sum and Difference of Angles Identities DISTANCE A geologist measures the angle between one side of a rectangular lot and the line from her position to the opposite corner of the lot as 30°. She then measures the angle between that line and the line to the point on the property where a river crosses as 45°. If z represents the distance between the upper left corner of the property and the point where the river crosses the top boundary, then Use the identity for tan(A – B) to find an exact value for z.

17. Example 2 Sum and Difference of Angles Identities Original equation UnderstandThe question asks for the distance between the upper left corner of the property to the line drawn along the river bank. PlanYou know the side adjacent to the 15° angle is 50 yards in length. SolveSolve for z. Difference identity

18. Example 2 Sum and Difference of Angles Identities Evaluate. Simplify and multiply each side by 50.

19. Example 2 Sum and Difference of Angles Identities Answer: CheckUse a calculator to find the Arctan of

20. Example 2 DISTANCE Suppose that in the previous example the distance along the river bank that is contained in the property is found to be 45 yards. If z still represents the distance between the upper left corner of the property and the point where the river crosses the top boundary, then holds true. Use the identity for sin(A – B) to find an exact value for z.

21. Example 2 A. B. C. D.

22. Example 3A Verify Trigonometric Identities A. Verify that the equation cos (360° –  ) = cos  is an identity. Answer: cos (360° –  )=cos  Original equation ? 1 ● cos  + 0 ● sin  =cos  Evaluate each expression. ? cos  =cos  Simplify. cos 360° ● cos  + sin 360° ● sin  =cos  ? Difference identity

23. Example 3B Verify Trigonometric Identities B. Verify that the equation cos (  –  ) = –cos  is an identity. Answer: cos (  –  )=–cos  Original equation ? –1 ● cos  + 0 ● sin  =–cos  Evaluate each expression. ? –cos  =–cos  Simplify. cos  ● cos  + sin  ● sin  =–cos  ? Difference identity

24. Example 3 A.yes; cos 90° cos θ – sin 90° sin θ = –sin θ B.yes; cos 90° sin θ – sin 90° cos θ = –sin θ C.yes; cos 90° sin θ + sin 90° cos θ = –sin θ D.no A. Determine whether cos (90° + θ) = –sin θ is an identity.

25. Example 3 A.yes; cos 180° sin θ – sin θ cos 90° = –sin θ B.yes; sin 180° cos θ + cos 180° sin θ = –sin θ C.yes; sin 180° cos θ – cos 180° sin θ = –sin θ D.no B. Determine whether sin (180° + θ) = –sin θ is an identity.

26. End of the Lesson