"Find the exact value of each expression. (tan)17π12" was answered by our experts

iohanetc

Answered question

2021-09-03

Find the exact value of each expression.

(\tan) \frac{17 π}{12}

Clara Reese

Skilled2021-09-04Added 120 answers

1. Split up the argument of the trigonometric function and then use the Sum formula for the Tangent function,

\tan (u + v) = \frac{\tan (u) + \tan (v)}{1 - \tan (u) \tan (v)}

, to find the value of given expression.
2.

\tan (\frac{17 π}{12}) = \tan (\frac{9 π}{12} + \frac{8 π}{12})

= \tan (\frac{3 π}{4} + \frac{2 π}{3})

= \frac{\tan (\frac{3 π}{4}) + \tan (\frac{2 π}{3})}{1 - (\tan (\frac{3 π}{4}) \tan (\frac{2 π}{3})}

= \frac{- 1 + (- \sqrt{3})}{1 - (- 1) \cdot (- \sqrt{3})}

= \frac{- 1 - \sqrt{3}}{1 - \sqrt{3}}

= \frac{(- 1 - \sqrt{3}) (1 + \sqrt{3})}{(1 - \sqrt{3}) (1 + \sqrt{3})}

= \frac{- 1 - 2 \sqrt{3} - 3}{1 - 3}

= \frac{4 - 2 \sqrt{3}}{- 2}

= 2 + \sqrt{3}

Result :

\tan (\frac{17 π}{12}) = 2 + \sqrt{3}

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