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Chapter 5: Problem 29

Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$\sin \frac{5 \pi}{12} \cos \frac{\pi}{4}-\cos \frac{5 \pi}{12} \sin \frac{\pi}{4}$$

Short Answer

Expert verified
The exact value of the given expression is \(\sqrt{3}/2\)

Step by step solution

01

Identify the expression type

In this expression, we see that its format is similar to the sine difference identity: \(\sin(A - B) = \sin A \cos B - \cos A \sin B\). So we can recognize this as the sine of the difference between two angles.
02

Apply the identity

Using the sine difference identity, our given expression \(\sin \frac{5 \pi}{12} \cos \frac{\pi}{4}-\cos \frac{5 \pi}{12} \sin \frac{\pi}{4}\) can be collapsed into \(\sin(\frac{5 \pi}{12}-\frac{\pi}{4})\)
03

Simplify the difference in the sine function

Let's simplify the difference inside the sine function, \(\frac{5 \pi}{12}-\frac{\pi}{4}\) = \(\frac{\pi}{3}\)
04

Calculate the value

We know that \(\sin \frac{\pi}{3} = \sqrt{3}/2\), as it's one of the special angles in trigonometric functions, we use the simplified angle to find the value of the expression

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Most popular questions from this chapter

Will help you prepare for the material covered in the first section of the next chapter. Solve each equation by using the cross-products principle to clear fractions from the proportion: If \(\frac{a}{b}=\frac{c}{d},\) then \(a d=b c,(b \neq 0 \text { and } d \neq 0)\) Round to the nearest tenth. $$\text { Solve for } a: \frac{a}{\sin 46^{\circ}}=\frac{56}{\sin 63^{\circ}}$$

Will help you prepare for the material covered in the next section. Use the appropriate values from Exercise 101 to answer each of the following. a. Is \(\cos \left(30^{\circ}+60^{\circ}\right),\) or \(\cos 90^{\circ},\) equal to \(\cos 30^{\circ}+\cos 60^{\circ} ?\) b. Is \(\cos \left(30^{\circ}+60^{\circ}\right),\) or \(\cos 90^{\circ},\) equal to \(\cos 30^{\circ} \cos 60^{\circ}-\sin 30^{\circ} \sin 60^{\circ} ?\)

Describe the difference between verifying a trigonometric identity and solving a trigonometric equation.

Determine whether each -statement makes sense or does not make sense, and explain your reasoning. I solved \(\cos \left(x-\frac{\pi}{3}\right)=-1\) by first applying the formula for the cosine of the difference of two angles.

Exercises \(116-118\) will help you prepare for the material covered in the next section. In each exercise, use exact values of trigonometric functions to show that the statement is true. Notice that each statement expresses the product of sines and/or cosines as a sum or a difference. $$\sin 60^{\circ} \sin 30^{\circ}=\frac{1}{2}\left[\cos \left(60^{\circ}-30^{\circ}\right)-\cos \left(60^{\circ}+30^{\circ}\right)\right]$$
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