Pre-Calc Exam Notes 77

Pre-Calc Exam Notes 77 - Sum and Difference Formulas...

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Sum and Difference Formulas Section 3.2 77 11. tan ( θ + 45 ) = 1 + tan θ 1 tan θ 12. cos ( A + B ) sin A cos B = cot A tan B 13. cot A + cot B = sin ( A + B ) sin A sin B 14. sin ( A B ) sin ( A + B ) = cot B cot A cot B + cot A 15. Generalize Exercise 6: For any a and b , r a 2 + b 2 a sin θ + b cos θ r a 2 + b 2 for all θ . 16. Continuing Example 3.12, use Snell’s law to show that the s-polarization transmission Fresnel coef±cient t 1 2 s = 2 n 1 cos θ 1 n 1 cos θ 1 + n 2 cos θ 2 (3.22) can be written as: t 1 2 s = 2 cos θ 1 sin θ 2 sin ( θ 2 + θ 1 ) x y 0 θ y = m 1 x + b 1 y = m 2 x + b 2 17. Suppose that two lines with slopes m 1 and m 2 , respectively, intersect at an angle θ and are not perpendicular (i.e. θ n= 90 ), as in the ±gure on the right. Show that tan θ = v v v v m 1 m 2 1 + m 1 m 2 v v v v . ( Hint: Use Example 1.26 from Section 1.5. ) 18. Use Exercise 17 to ±nd the angle between the lines y = 2 x + 3 and y =− 5 x 4. 19. For any triangle ABC , show that cot A cot B + cot B cot C + cot C cot A = 1. ( Hint: Use Exercise 9 and C = 180 ( A + B ) . ) 20. For any positive angles A , B , and C such that
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