Pre-Calc Exam Notes 86

Pre-Calc Exam Notes 86 - A + sin 2 B + sin 2 C = 4 sin A...

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86 Chapter 3 Identities §3.4 Exercises 1. Prove formula (3.38). 2. Prove formula (3.39). 3. Prove formula (3.40). 4. Prove formula (3.41). 5. Prove formula (3.42). 6. Prove formula (3.44). 7. Prove Mollweide’s second equation: For any triangle ABC , a + b c = cos 1 2 ( A B ) sin 1 2 C . 8. Continuing Example 3.21, use Snell’s law to show that the p-polarization refection Fresnel coe±²- cient r 1 2 p = n 2 cos θ 1 n 1 cos θ 2 n 2 cos θ 1 + n 1 cos θ 2 (3.46) can be written as: r 1 2 p = tan ( θ 1 θ 2 ) tan ( θ 1 + θ 2 ) 9. There is a more general form for the instantaneous power p ( t ) = v ( t ) i ( t ) in an electrical circuit than the one in Example 3.22. The voltage v ( t ) and current i ( t ) can be given by v ( t ) = V m cos ( ω t + θ ) , i ( t ) = I m cos ( ω t + φ ) , where θ is called the phase angle . 4 Show that p ( t ) can be written as p ( t ) = 1 2 V m I m cos ( θ φ ) + 1 2 V m I m cos (2 ω t + θ + φ ) . For Exercises 10-15, prove the given identity or inequality for any triangle ABC . 10. sin A + sin B + sin C = 4 cos 1 2 A cos 1 2 B cos 1 2 C ( Hint: Mimic Example 3.18 using (sin A + sin B ) + (sin C sin ( A + B + C )) . ) 11. cos A + cos ( B C ) = 2 sin B sin C 12. sin 2
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Unformatted text preview: A + sin 2 B + sin 2 C = 4 sin A sin B sin C ( Hints: Group sin 2 B and sin 2 C together, use the double-angle ±ormula ±or sin 2 A, use Exercise 11. ) 13. a − b a + b = sin A − sin B sin A + sin B 14. cos 1 2 A = r s ( s − a ) bc and sin 1 2 A = r ( s − b ) ( s − c ) bc , where s = 1 2 ( a + b + c ) ( Hint: Use the Law o± Cosines to show that 2 bc (1 + cos A ) = 2 s ( s − a ) . ) 15. 1 2 (sin A + sin B ) ≤ sin 1 2 ( A + B ) ( Hint: Show that sin 1 2 ( A + B ) − 1 2 (sin A + sin B ) ≥ . ) 16. In Example 3.20, which angles A , B , C give the maximum value of cos A + cos B + cos C ? 4 Though it does not matter for this exercise, none of the angles in these formulas are measured in degrees. We will discuss their unit of measurement in Chapter 4....
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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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