CK- 12 Algebra II with Trigonometry Concepts 1
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1 14.1 Graphing Sine and Cosine 1. A.,1 B. (, 1) C. 3,0 D. 11 1, 6 E. (, 1) F. G. H. 11, 4 7, 1 11, ,,,,,,, , and, 3 3 CK- 1 Algebra II with Trigonometry Concepts 1
2 4.ans ans ans ans ans ans CK- 1 Algebra II with Trigonometry Concepts
3 10.ans ans CK- 1 Algebra II with Trigonometry Concepts 3
4 1.ans units 14. units 15. y =.5sinx 16. y = 1.75cosx CK- 1 Algebra II with Trigonometry Concepts 4
5 14. Translating Sine and Cosine Functions 1. C. A 3. D 4. B 5. C 6. D 5 7. y = sin x Infinitely many. Explanations will vary. 9. a) b) 10.ans ans CK- 1 Algebra II with Trigonometry Concepts 5
6 1.ans ans ans ans Yes, explanations will vary. CK- 1 Algebra II with Trigonometry Concepts 6
7 14.3 Putting it all Together 1. T. T 3. F 4. F 5. T 6.ans ans D:, R: y [, 0] D:, R: y [5, 1] CK- 1 Algebra II with Trigonometry Concepts 7
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8 8.ans ans D:, R: 3 3 y, 4 4 D:, R: y [3, 7] 10.ans ans D:, R: y [0.5, 3.5] D:, R: y [6.8, 1.] CK- 1 Algebra II with Trigonometry Concepts 8
9 1. y =.5sinx y =.5cos x every ( x ) y =.5sin y =.5cos x y = sin x y ( x ) = cos y = sin x y =cosx 1 0. will vary. CK- 1 Algebra II with Trigonometry Concepts 9
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10 14.4 Changes in the Period of a Sine and Cosine Function D:, R: y [5, 5] ans minimums: maximums: 6 3 n, 5 ± 3 n, 5 ± CK- 1 Algebra II with Trigonometry Concepts 10
11 x = 0,,,,,, D:, R: y 1, 1 ans minimums: 8 8, 1 and (0, 1) 3 3 n ± 1. x =, 3 maximums: n, 1 ± 13. D:, R: y [3, 3] ans CK- 1 Algebra II with Trigonometry Concepts 11
12 14. minimums: 3 ± n, 3 4 maximums: ± n, x = 0,,,, 16. D:, R: y a + k, a + k y = sin x y = 3 sin 5 5 x 19. y = 9sin x x =,,,, CK- 1 Algebra II with Trigonometry Concepts 1
13 14.5 Graphing Tangent *n is any integer. 1. p =, D: ; x ± n, R:. p =, D: ; x ± n, R: ans ans- 3. p =, D: ; x ± n, R: 4. p =, D: ; x ± n, R: ans ans CK- 1 Algebra II with Trigonometry Concepts 13
14 5. p =, D: ; x ± n, R: p =, D: ; x ± n, R: ans ans p =, D: ; x ± n, R: 8. p =, D: ; x ± n, R: ans ans- CK- 1 Algebra II with Trigonometry Concepts 14
15 9. p =, D: ; x ± n, R: 10. x =± n x =± n 3 x = n 4 y = 3tan x y = 1 1 tan y =.5tan x p = 3, D: ; x ± 3 n, 4 x CK- 1 Algebra II with Trigonometry Concepts 15
16 14.6 Introduction to Trig Identities 1. Student needs to show proof.. Student needs to show proof. 3. Student needs to show proof. 4. The graphs overlap. 5. Student needs to show proof. 6. Hint: Sine is odd and cosine is even. 7. Hint: Change everything to sine and cosine. 8. Hint: Change everything to sine and cosine. 9. Hint: Change cosecant using the Reciprocal Identity. 10. Hint: Change cotangent to tangent using the Reciprocal Identity. 11. Hint: Change everything to sine and cosine. 1. Hint: Use the Negative Angle Identity for sine. 13. Hint: Plug in value for θ into the Pythagorean Identity. 14. Hint: Plug in value for θ into the Pythagorean Identity. 15. Hint: Plug in value for θ into the Pythagorean Identity. 16. Odd: Sine, Tangent, Cosecant, Cotangent. Even: Cosine, Secant. CK- 1 Algebra II with Trigonometry Concepts 16
17 14.7 Using Identities to Find Exact Trigonometric Values 1. I and II. III and IV.. I and IV. II and III. 3. I and III. II and IV cos θ =, tan θ =, csc θ =, sec θ =, cotθ = sin θ = , tan θ =, csc θ =, sec θ =, cotθ = cos θ =, sin θ =, csc θ =, sec θ =, cotθ = sin θ =, cos θ =, tan θ =, csc θ =, cotθ = cos θ =, tan θ =, csc θ =, sec θ =, cotθ = sin θ =, tanθ = 1, cscθ =, secθ =, cotθ = sin θ =, cos θ =, tan θ =, sec θ =, cscθ = sin θ =, cos θ =, tan θ =, sec θ =, cotθ = cos θ =, sin θ =, csc θ =, sec θ =, cotθ = The Pythagorean Theorem CK- 1 Algebra II with Trigonometry Concepts 17
18 14.8 Simplifying Trigonometric Expressions 1. cosx. cosxsinx 3. cot x 4. cos x 5. cscx sin x cos x sec x csc x 11. sinx 1. tanx 13. cos x secx CK- 1 Algebra II with Trigonometry Concepts 18
19 14.9 Verifying a Trigonometric Identity 1. Hint: Use the Reciprocal Identities.. Hint: Use the Reciprocal Identities. 3. Hint: Change everything to sine and cosine. 4. Hint: Change everything to sine and cosine. 5. Hint: Use the Cofunction Identities. 6. Hint: Use the Cofunction Identities. 7. Hint: Change everything into sine and cosine. 8. Hint: Use the Pythagorean Identities. 9. Hint: FOIL. 10. Hint: Combine like terms. 11. Hint: Start with the Pythagorean Identities. 1. Hint: Change right hand side into terms of sine and cosine. 13. Hint: Find a common denominator for the left hand side. 14. Hint: Use the Pythagorean Identities. 15. Hint: Change left hand side into terms of sine and cosine. You may also need to find a common denominator and/or FOIL. CK- 1 Algebra II with Trigonometry Concepts 19
20 14.10 Solving Trigonometric Equations using Algebra *n is any integer. 1. yes. no 3. yes 4. x = n x = ± n 6 5 x = ± n, ± n no solution x = ± n, ± n 3 3 x = ± n, where n is not a multiple of x =, x =, no solution 13. x = 0.775, x =,,, x =,,, CK- 1 Algebra II with Trigonometry Concepts 0
21 14.11 Solving Trigonometric Equations using Quadratic Techniques x =,, x =,3.3943, x = 0,, x =,, x =,3.4814, x = 0.57,, x = 0,,, 3 7 x =, x = ,1.983, 10. x = 11. x = x =, x =,,,,, (0.3919, ), (.7497, ) 15. (4.1461, ), (5.534, 1.9) CK- 1 Algebra II with Trigonometry Concepts 1
22 14.1 Finding Exact Trig Values using Sum and Difference Formulas Yes will vary Any combination that adds up to 14 will work. 15. Students must provide proof. CK- 1 Algebra II with Trigonometry Concepts
23 14.13 Simplifying Trig Expressions using Sum and Difference Formulas sinx 8. cosx 9. cosx 10. sinx 11. tanx 1. tanx ( cos x 3sin x) tanx 1 tanx ( cos x + 3sin x) 16. F 17. T 18. F CK- 1 Algebra II with Trigonometry Concepts 3
24 14.14 Solving Trig Equations using Sum and Difference Formulas 1. 3 x =, 4 4. x = 3. x = 0, x =, x =, x = 0, 7. x = 8. x = x = 0,,, x = 0, 11. no solution x =, 4 4 x = x = At 5.7 sec and 1.14 min. CK- 1 Algebra II with Trigonometry Concepts 4
25 14.15 Finding Exact Trig Values using Double and Half Angle Formulas CK- 1 Algebra II with Trigonometry Concepts 5
26 CK- 1 Algebra II with Trigonometry Concepts 6
27 14.16 Simplifying Trig Expressions using Double and Half Angle Formulas cosx sec x sinx cosx sinx 1 5sin x 5. sinx 6. sin x(1 + cos x) x 1 7. Hint: Change cot to x or tan x cos x sin 8. Hint: Cross-multiply. 9. Hint: Expand sinx and cosx 10. Hint: FOIL 11. Hint: Rewrite the half-angles 1. Hint: Rewrite cscx in terms of sine 13. Hint: cos3x = cos( x + x) 14. Hint: Use cos cos sin x = x x 15. Hint: Expand the double-angles 16. Hint: Factor CK- 1 Algebra II with Trigonometry Concepts 7
28 14.17 Solving Trig Equations using Double and Half Angle Formulas x = 0,,, x = 0,,,,, x = 0,,, 4. x = 0, 5. x =.37, x =, 7. x = x =,, x = 0,,, x = 0, 11. no solution 1. x = 0, x = 0,, x =,,, no solution 16. infinitely many solutions CK- 1 Algebra II with Trigonometry Concepts 8
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