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Chapter 5: Problem 22

Find the exact value of each expression. $$\tan \left(\frac{\pi}{3}+\frac{\pi}{4}\right)$$

Short Answer

Expert verified
The exact value of the expression \(\tan \left(\frac{\pi}{3}+\frac{\pi}{4}\right)\) is \(-1 - \sqrt{3}\)

Step by step solution

01

Recognize the formula

Note that we have a trigonometric function that involves the sum of two angles, so we should use the formula for \(\tan(a+b)\), which is \(\frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}\)
02

Apply the formula to our specific problem

Let's replace \(a\) with \(\frac{\pi}{3}\), and \(b\) with \(\frac{\pi}{4}\) in our formula. This gives us \(\frac{\tan(\frac{\pi}{3}) + \tan(\frac{\pi}{4})}{1 - \tan(\frac{\pi}{3})\tan(\frac{\pi}{4})}\)
03

Replace with known values of tan

We know that \(\tan(\frac{\pi}{3}) = \sqrt{3}\) and \(\tan(\frac{\pi}{4}) = 1\). Plugging these values into the equation we derived in Step 2, we get \(\frac{\sqrt{3} + 1}{1-\sqrt{3} \cdot 1}\).
04

Simplify the expression

After performing the operations, we can simplify the expression to \(\frac{\sqrt{3} + 1}{1 - \sqrt{3}} = -1 - \sqrt{3}\)

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Most popular questions from this chapter

Graph: \( f(x)=\frac{5 x^{2}}{x^{2}-25}\) (Section \(2.6, \text { Example } 6)\)

Solve: \(\log x+\log (x+1)=\log 12\) (Section 3.4, Example 8)

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Solve each equation on the interval \([0,2 \pi)\) Do not use a calculator. $$\sin 3 x+\sin x+\cos x=0$$

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