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Chapter 5: Problem 41

Find the exact value of the expression. $$\frac{\tan (5 \pi / 6)-\tan (\pi / 6)}{1+\tan (5 \pi / 6) \tan (\pi / 6)}$$

Short Answer

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

Step by step solution

01

Identify the formula

Recognize that the given expression fits into the format of the tangent subtraction formula, \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \]. Here, \(a = 5\pi/6\) and \(b = \pi/6\)
02

Apply the Tangent Subtraction Rule

The given expression can thus be written in the form of \(\tan(a - b)\). Plugging in the values of \(a\) and \(b\), we get \(\tan((5\pi / 6) - (\pi / 6))\)
03

Simplify the expression

Subtract the two fractions to get the new angle for the tangent function: \( \tan(4\pi / 6) \)
04

Simplify further

Simplifying further, \(\tan(4\pi / 6)\) equals to \(\tan(2\pi / 3)\) as \(4 / 6\) simplifies to \(2 / 3 \).
05

Compute the trigonometric value

Lastly, using the unit circle or trigonometric table, compute the value of \(\tan(2\pi/3)\). Since \(\tan(2\pi/3)\) equals to \(-\sqrt{3}\), this is the final result

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