Pre-Calc Exam Notes 81

Pre-Calc Exam Notes 81 - 1 2 θ sin θ tan 1 2 θ 360 ◦...

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Double-Angle and Half-Angle Formulas Section 3.3 81 Example 3.15 Prove the identity sec 2 1 2 θ = 2 sec θ sec θ + 1 . Solution: Since secant is the reciprocal of cosine, taking the reciprocal of formula (3.29) for cos 2 1 2 θ gives us sec 2 1 2 θ = 2 1 + cos θ = 2 1 + cos θ · sec θ sec θ = 2 sec θ sec θ + 1 . Exercises For Exercises 1-8, prove the given identity. 1. cos 3 θ = 4 cos 3 θ 3 cos θ 2. tan 1 2 θ = csc θ cot θ 3. sin 2 θ sin θ cos 2 θ cos θ = sec θ 4. sin 3 θ sin θ cos 3 θ cos θ = 2 5. tan 2 θ = 2 cot θ tan θ 6. tan 3 θ = 3 tan θ tan 3 θ 1 3 tan 2 θ 7. tan 2 1 2 θ = tan θ sin θ tan θ + sin θ 8. cos 2 ψ cos 2 θ = 1 + cos 2 ψ 1 + cos 2 θ 9. Some trigonometry textbooks used to claim incorrectly that sin θ + cos θ = r 1 + sin 2 θ was an identity. Give an example of a speci±c angle θ that would make that equation false. Is sin θ + cos θ = ± r 1 + sin 2 θ an identity? Justify your answer. 10. Fill out the rest of the table below for the angles 0 < θ < 720 in increments of 90 , showing θ , 1 2 θ , and the signs ( + or ) of sin θ and tan 1 2 θ . θ 1 2 θ sin θ tan 1 2 θ 0 90 0 45 + + 90 180 45 90 180 270 90 135 270 360 135 180 θ
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Unformatted text preview: 1 2 θ sin θ tan 1 2 θ 360 ◦ − 450 ◦ 180 ◦ − 225 ◦ 450 ◦ − 540 ◦ 225 ◦ − 270 ◦ 540 ◦ − 630 ◦ 270 ◦ − 315 ◦ 630 ◦ − 720 ◦ 315 ◦ − 360 ◦ 11. In general, what is the largest value that sin θ cos θ can take? Justify your answer. For Exercises 12-17, prove the given identity for any right triangle △ ABC with C = 90 ◦ . 12. sin ( A − B ) = cos 2 B 13. cos ( A − B ) = sin 2 A 14. sin 2 A = 2 ab c 2 15. cos 2 A = b 2 − a 2 c 2 16. tan 2 A = 2 ab b 2 − a 2 17. tan 1 2 A = c − b a = a c + b 18. Continuing Exercise 20 from Section 3.1, it can be shown that r (1 − cos θ ) = a (1 + ǫ )(1 − cos ψ ) , and r (1 + cos θ ) = a (1 − ǫ )(1 + cos ψ ) , where θ and ψ are always in the same quadrant. Show that tan 1 2 θ = r 1 + ǫ 1 − ǫ tan 1 2 ψ ....
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.

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