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Chapter 5: Problem 31

Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$\frac{\tan \frac{\pi}{5}-\tan \frac{\pi}{30}}{1+\tan \frac{\pi}{5} \tan \frac{\pi}{30}}$$

Short Answer

Expert verified
The given expression is equivalent to \(\tan \frac{\pi}{6}\) whose value is \(\frac{\sqrt{3}}{3}\)

Step by step solution

01

Identify the Formula

The expression resembles the formula for the tangent of the difference of two angles. It is formatted as \(\frac{\tan a - \tan b}{1 + \tan a \tan b}\), which is equivalent to \(\tan(a-b)\).
02

Apply the Formula

Substituting \(\frac{\pi}{5}\) for \(a\) and \(\frac{\pi}{30}\) for \(b\) into formula, the given expression equals \(\tan (\frac{\pi}{5} - \frac{\pi}{30})\)
03

Simplify the Expression

Subtract the angles in the parentheses to find the angle that the expression is equivalent to. This gives \(\tan (\frac{6\pi}{30} - \frac{\pi}{30}) = \tan \frac{5\pi}{30}\)
04

Convert to Simplified Form

Simplify \(\frac{5\pi}{30}\) to \(\frac{\pi}{6}\). So the expression is equal to \(\tan \frac{\pi}{6}\)
05

Find the Value

Find the value of \(\tan \frac{\pi}{6}\) to get the exact value of the expression. This is readily available in trigonometric tables or can be calculated as \(\frac{\sqrt{3}}{3}\)

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Most popular questions from this chapter

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$2 \tan ^{2} x+5 \tan x+3=0$$

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$\sin \left(x+\frac{\pi}{4}\right)=\sin x+\sin \frac{\pi}{4}$$

Determine the amplitude, period, and phase shift of \(y=4 \sin (2 \pi x+2) .\) Then graph one period of the function. (Section 4.5, Example 4 )

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation \(\tan x=\frac{\pi}{2}\) has no solution.

Verify each identity. $$\ln e^{\tan ^{2} x-\sec ^{2} x}=-1$$
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