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Splash Screen. Five-Minute Check (over Lesson 5-1) Then/Now New Vocabulary Example 1: Verify a Trigonometric Identity Example 2: Verify a Trigonometric Identity by Combining Fractions Example 3: Verify a Trigonometric Identity by Multiplying

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 5-1) Then/Now New Vocabulary Example 1: Verify a Trigonometric Identity Example 2: Verify a Trigonometric Identity by Combining Fractions Example 3: Verify a Trigonometric Identity by Multiplying Example 4: Verify a Trigonometric Identity by Factoring Example 5: Verify an Identity by Working Each Side Separately Concept Summary: Strategies for Verifying Trigonometric Identities Example 6: Determine Whether an Equation is an Identity Lesson Menu

  3. Find the value of the expression using the given information. If tan θ = , find cot θ. A. B. C. D. 5–Minute Check 1

  4. Find the value of the expression using the given information. If sin θ = and cos θ = , find tan θ. A. B. C. D. 5–Minute Check 2

  5. A. B. C. D. Find the value of the expression using the given information. If csc θ = 3 and cos θ < 0, find cos θ and tan θ. 5–Minute Check 3

  6. Simplify csc x – csc x cos2x. A.sin x B.cos x C.csc x(1 + sin x) D.1 – cos x 5–Minute Check 4

  7. If sin θ = 0.59, find . A.−0.81 B.−0.59 C.0.59 D.0.81 5–Minute Check 5

  8. You simplified trigonometric expressions. (Lesson 5-1) • Verify trigonometric identities. • Determine whether equations are identities. Then/Now

  9. verify an identity Vocabulary

  10. Verify that . Pythagorean Identity Reciprocal Identity Simplify.  Verify a Trigonometric Identity The left-hand side of this identity is more complicated, so transform that expression into the one on the right. Example 1

  11. Answer: Verify a Trigonometric Identity Example 1

  12. Verify that 2 – cos2x = 1 + sin2x. A. 2 – cos2x = –(sin2x + 1) + 2 = 1 + sin2x B. 2 – cos2x = 2 – (sin2x + 1) = 1 + sin2x C. 2 – cos2x = 2 – (1 + sin2x) + 2 = 1 + sin2x D. 2 – cos2x = 2 – (1 – sin2x) = 1 + sin2x Example 1

  13. Verify that . Start with the right hand side of the identity. Common denominator Distributive Property Verify a Trigonometric Identity by Combining Fractions The right-hand side of the identity is more complicated, so start there, rewriting each fraction using the common denominator 1 – cos2x. Example 2

  14. Simplify. Divide out the common factor of sin x. Simplify. Quotient Identity  Verify a Trigonometric Identity by Combining Fractions Example 2

  15. Verify a Trigonometric Identity by Combining Fractions Answer: Example 2

  16. Verify that . A. B. C. D. Example 2

  17. Verify that . Multiply the numerator and denominator by the conjugate of sec x – 1, which is sec x + 1. Multiply. Verify a Trigonometric Identity by Multiplying Because the left-hand side of this identity involves a fraction, it is slightly more complicated than the right side. So, start with the left side. Example 3

  18. Pythagorean Identity Quotient Identity Multiply by the reciprocal of the denominator. Divide out the common factor of sin x. Verify a Trigonometric Identity by Multiplying Example 3

  19. Distributive Property Rewrite the fraction as the sum of two fractions; Reciprocal Identity. Divide out the common factor of cos x. Verify a Trigonometric Identity by Multiplying Example 3

  20. Quotient Identity  Verify a Trigonometric Identity by Multiplying Example 3

  21. Verify a Trigonometric Identity by Multiplying Answer: Example 3

  22. Verify that . A. B. C. D. Example 3

  23. Start with the left-hand side of the identity. Factor. cos x sec2x tan x –cos x tan3x= cos x tan x (sec2x –tan2x) = cos x tan x (1) Pythagorean Identity Quotient Identity = Divide out the common factor of cos x. = sin x  Verify a Trigonometric Identity by Factoring Verify that cos x sec2x tan x – cos x tan3x = sin x. Example 4

  24. Answer:cos x sec2x tan x –cos x tan3x = cos x tan x (sec2x –tan2x) = cos x tan x (1) = = sin x Verify a Trigonometric Identity by Factoring Example 4

  25. A. B. C. D. Verify that csc x – cos x csc x – cos x cot x + cot x = sin x. Example 4

  26. Verify that . Write as the sum of two fractions. Simplify and apply a Reciprocal Identity. Verify an Identity by Working Each Side Separately Both sides look complicated, but there is a clear first step for the expression on the left. So, start with the expression on the left. Example 5

  27. From here, it is unclear how to transform 1 + cot x into , so start with the right side and work to transform it into the intermediate form 1 + cot x. Pythagorean Identity Simplify. Factor. Verify an Identity by Working Each Side Separately Example 5

  28. Divide out the common factor of 1 – cot x. Write as the sum of two fractions. Simplify and apply a Reciprocal Identity. Multiply by . Verify an Identity by Working Each Side Separately To complete the proof, work backward to connect the two parts of the proof. Example 5

  29. Simplify. Pythagorean Identity Simplify.  Answer: Verify an Identity by Working Each Side Separately Example 5

  30. A. B. C. D. Verify that tan2x – sin2x = sin2x tan2x. Example 5

  31. Key Concept 6

  32. A. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal. Determine Whether an Equation is an Identity The equation appears to be an identity because the graphs of the related functions over [–2π, 2π] scl: π by [–1, 3] scl: 1 coincide. Verify this algebraically. Example 6

  33. Pythagorean Identity Divide out the common factor of sec x. Determine Whether an Equation is an Identity Example 6

  34. Reciprocal Identities Simplify. Quotient Identity Determine Whether an Equation is an Identity Answer: Example 6

  35. B. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal. Determine Whether an Equation is an Identity Example 6

  36. The graphs of the related functions do not coincide for all values of x for which the both functions are defined. When , Y1 ≈ 1.43 but Y2 ≈ –0.5. The equation is not an identity. Determine Whether an Equation is an Identity Example 6

  37. Answer:When , Y1 ≈ 1.43 but Y2 = –0.5. The equation is not an identity. Determine Whether an Equation is an Identity Example 6

  38. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find a value for which both sides are defined but not equal. A. The equation appears to be an identity because the graphs of the related functions over [–2π, 2π] scl: π by [–3, 3] scl: 1 coincide. B. When , Y1 ≈ 0.71 but Y2 ≈ 0.29. The equation is not an identity. Example 6

  39. End of the Lesson

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