Pre-Calc Exam Notes 76

Pre-Calc Exam Notes 76 - . 5. Use 15 = 45 30 to nd the...

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76 Chapter 3 Identities §3.2 Solution: Multiply the top and bottom of r 1 2 s by sin θ 1 sin θ 2 to get: r 1 2 s = n 1 cos θ 1 n 2 cos θ 2 n 1 cos θ 1 + n 2 cos θ 2 · sin θ 1 sin θ 2 sin θ 1 sin θ 2 = ( n 1 sin θ 1 ) sin θ 2 cos θ 1 ( n 2 sin θ 2 ) cos θ 2 sin θ 1 ( n 1 sin θ 1 ) sin θ 2 cos θ 1 + ( n 2 sin θ 2 ) cos θ 2 sin θ 1 = ( n 1 sin θ 1 ) sin θ 2 cos θ 1 ( n 1 sin θ 1 ) cos θ 2 sin θ 1 ( n 1 sin θ 1 ) sin θ 2 cos θ 1 + ( n 1 sin θ 1 ) cos θ 2 sin θ 1 (by Snell’s law) = sin θ 2 cos θ 1 cos θ 2 sin θ 1 sin θ 2 cos θ 1 + cos θ 2 sin θ 1 = sin ( θ 2 θ 1 ) sin ( θ 2 + θ 1 ) The last two examples demonstrate an important aspect of how identities are used in practice: recognizing terms which are part of known identities, so that they can be factored out. This is a common technique. Exercises 1. Verify the addition formulas (3.12) and (3.13) for A = B = 0 . For Exercises 2 and 3, ±nd the exact values of sin ( A + B ), cos ( A + B ), and tan ( A + B ). 2. sin A = 8 17 , cos A = 15 17 , sin B = 24 25 , cos B = 7 25 3. sin A = 40 41 , cos A = 9 41 , sin B = 20 29 , cos B = 21 29 4. Use 75 = 45 + 30 to ±nd the exact value of sin 75
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Unformatted text preview: . 5. Use 15 = 45 30 to nd the exact value of tan 15 . 6. Prove the identity sin + cos = r 2 sin ( + 45 ) . Explain why this shows that r 2 sin + cos r 2 for all angles . For which between 0 and 360 would sin + cos be the largest? For Exercises 7-14, prove the given identity. 7. cos ( A + B + C ) = cos A cos B cos C cos A sin B sin C sin A cos B sin C sin A sin B cos C 8. tan ( A + B + C ) = tan A + tan B + tan C tan A tan B tan C 1 tan B tan C tan A tan C tan A tan B 9. cot ( A + B ) = cot A cot B 1 cot A + cot B 10. cot ( A B ) = cot A cot B + 1 cot B cot A...
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