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Splash Screen. Five-Minute Check (over Chapter 4) Then/Now New Vocabulary Key Concept: Reciprocal and Quotient Identities Example 1: Use Reciprocal and Quotient Identities Key Concept: Pythagorean Identities Example 2: Use Pythagorean Identities Key Concept: Cofunction Identities

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  1. Splash Screen

  2. Five-Minute Check (over Chapter 4) Then/Now New Vocabulary Key Concept: Reciprocal and Quotient Identities Example 1: Use Reciprocal and Quotient Identities Key Concept: Pythagorean Identities Example 2: Use Pythagorean Identities Key Concept: Cofunction Identities Key Concept: Odd-Even Identities Example 3: Use Cofunction and Odd-Even Identities Example 4: Simplify by Rewriting Using Only Sine and Cosine Example 5: Simplify by Factoring Example 6: Simplify by Combining Fractions Example 7: Rewrite to Eliminate Fractions Lesson Menu

  3. A. B. C. D. Find the exact values of the six trigonometric functions of θ. 5–Minute Check 1

  4. If , find the exact values of the fiveremaining trigonometric function values of θ. A. B. C. D. 5–Minute Check 2

  5. A. B. C. D. Write −150° in radians as a multiple of π. 5–Minute Check 3

  6. Solve ∆ABC if A = 33°, b = 9, and c = 13.Round side lengths to the nearest tenth andangle measures to the nearest degree. A.B = 42°, C = 105°, a = 7.3 B.B = 52°, C = 95°, a = 9 C.B = 95°, C = 52°, a = 7.3 D.B = 105°, C = 42°, a = 7.3 5–Minute Check 4

  7. Find the exact value of A. B. C. D. 5–Minute Check 5

  8. You found trigonometric values using the unit circle. (Lesson 4-3) • Identify and use basic trigonometric identities to find trigonometric values. • Use basic trigonometric identities to simplify and rewrite trigonometric expressions. Then/Now

  9. identity • trigonometric identity • cofunction • odd-even identities Vocabulary

  10. Key Concept 1

  11. A. If , find sec θ. Reciprocal Identity Divide. Answer: Use Reciprocal and Quotient Identities Example 1

  12. B. If and , find sin x. Reciprocal Identity Quotient Identity Substitute for cos x. Use Reciprocal and Quotient Identities Example 1

  13. Divide. Multiply each side by . Simplify. Answer: Use Reciprocal and Quotient Identities Example 1

  14. If , find sin . A. B. C. D. Example 1

  15. Key Concept 2

  16. = csc θTake the square root of each side. Reciprocal Identity Solve for sin θ. Use Pythagorean Identities If cot θ = 2 and cos θ < 0, find sin θ and cos θ. Use the Pythagorean Identity that involves cot θ. cot2θ + 1 = csc2θ Pythagorean Identity (2)2+ 1 = csc2θ cot θ = 2 5 = csc2θ Simplify. Example 2

  17. Since is positive and cos θ < 0, sin θ must be negative. So . You can then use this quotient identity again to find cos θ. Quotient Identity cot θ = 2 and Multiply each side by . Use Pythagorean Identities Example 2

  18. So, Answer: Simplify.  Use Pythagorean Identities Check sin 2θ + cos 2θ = 1 Pythagorean Identity Example 2

  19. Find the value of csc  and cot  if and cos < 0. A. B. C. D. Example 2

  20. Key Concept 3

  21. Key Concept 4

  22. If cos x = –0.75, find Factor. Odd-Even Identity Cofunction Identity cos x = –0.75 Simplify. Use Cofunction and Odd-Even Identities Example 3

  23. So, = 0.75. Use Cofunction and Odd-Even Identities Answer:0.75 Example 3

  24. If cos x = 0.73, find . A. –0.73 B. –0.68 C. 0.68 D. 0.73 Example 3

  25. So, = cos x. Simplify . Pythagorean Identity Multiply. Simplify. Simplify by Rewriting Using Only Sine and Cosine Solve Algebraically Example 4

  26. The graphs of and y = cos x appear to be identical. Simplify by Rewriting Using Only Sine and Cosine Support Graphically Answer:cos x Example 4

  27. Simplify csc x – cos x cot x. A. cot x B. tan x C. cos x D. sin x Example 4

  28. Pythagorean Identity Quotient Identity Multiply. Factor. = sin3x Simplify. Simplify by Factoring Simplify cos x tan x – sin x cos2x. Solve Algebraically cos x tan x – sin x cos2x Original expression So, cos x tan x –sin x cos2x = sin3x. Example 5

  29. Simplify by Factoring Support Graphically The graphs below appear to be identical. Answer:sin3x Example 5

  30. Simplify cos2x sin x – cos(90° – x). A. –sin3x B. sin3x C. cos2x – 1 D. sin x cos x Example 5

  31. Simplify . Common denominator Multiply. Add the numerators. Simplify. Pythagorean Identity Simplify by Combining Fractions Example 6

  32. Reciprocal Identity Reciprocal and Quotient Identities Divide out common factor. Reciprocal Identity –2csc2 x. Simplify by Combining Fractions Answer:– 2 sec2x – 2 sec2x Example 6

  33. Simplify . A. cos x B. 2 + 2 cos x C. 2 sin x D. 2 csc x Example 6

  34. Rewrite as an expression that does not involve a fraction. Pythagorean Identity Reciprocal Identity Reciprocal Identity Quotient Identity Rewrite to Eliminate Fractions Example 7

  35. So, = tan2 x. Rewrite to Eliminate Fractions Answer:tan2x Example 7

  36. Rewrite as an expression that does not involve a fraction. A. –2 tan2 x B. 1+ sin x C. 1 – cos x D. 2 sin x Example 7

  37. End of the Lesson

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