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# Solving Trigonometry functions

1. Find the exact value of the sine, cosine, and tangent of the angle. 11pie/12

2. Find all solutions of the equation in the interval [0, 2pie).
Sin(x-5pie/6)-sin(x+ 5pie/6) =1

3. Find the exact value of sin2x using the double angle formula. Please help explain the use of the double angle formula.
Sin x=1/7, 0<x<pie/2

4. Given cos theta = 4/9, where 3pie/2 <or equal to theta <or equal to 2pie.

5. Express 2sin3xcos6x as a sum containing only sines or cosines.

6. Express cos7x-cos5x as a product containing only sines and/or cosines.

7. Evaluate the expressions without the aid of a calculator.

a. Arctan(- sqrt3/3)
b. Arcsin(-1/2)

8. Use inverse functions to evaluate the expressions,

a. cos(arcsin(sqrt5/5))
b. cos(arctan(sqrt2/x))

9. Identify the x-values that are solutions of the equations.

a. 8cos x-4 = 0
b. 18cot^2 x-18 = 0

10. A large pole is 175 feet tall. On a particular day at noon it casts a 198-foot shadow. What is the sun's angle of elevation?

#### Solution Preview

Following is the text part of the solution. Please see the attached file for complete solution. Equations, diagrams, graphs and special characters will not appear correctly here. Thank You.
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You need to refresh all your trig identities. There are plenty of web sites which talk about those. I have high lighted the trig identities that I used in each problem.

1. Find the exact value of the sine, cosine, and tangent of the angle. 11pie/12

Use : sin x = sin (p-x)

Sin (11p/12) = sin(p-11p/12) = sin(p/12)

Since we know what is sin pi/6 using the following identity makes sense.

Use: cos 2x = 1 - 2sin2 x è sinx = sqrt {½ (1-cos2x)}

Sin (11p/12) = sin(p-11p/12) = sin(p/12) = sqrt {½ (1-cos(p/6)}= Ö (1/2 (1-Ö3/2))
Sin (11p/12 = Ö(2-Ö3)/2 This is the exact value.

Evaluating we get Sin (11p/12) = 0.258819

2. Find all solutions of the equation in the interval [0, 2pie).
Sin(x-5pie/6)-sin(x+ 5pie/6) =1

Use the trig identity : sin C - sin D = 2 cos (C+D)/2 * sin (C-D)/2

Sin(x-5pie/6)-sin(x+ 5pie/6) = 2 cos x * sin -(5pi/6) = 1

2 cos x * sin -(5pi/6) = 1 è cos x * sin 5pi/6 = - 1/2,
we know sin 5pi/6 = sin pi - 5pi/6 = sin pi/6 = ½

cos x * 1/2 = -1/2, è cos x = -1

Between 0 and 2pi we have only one ...

#### Solution Summary

I have provided step by step solutions to all the questions. Every trigonometry identity used has been typed out. This is a very good question and answer set on solving trig identities. Great practice questions as well.

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