Question

find the exact value of the expression. do not use a calculator.
2 sin \\( \frac { \pi } { 3 } - 6 \tan \frac { \pi } { 6 } \\)
select the correct choice below and, if necessary, fill in the answer box to complete your choice
a. \\( 2 \sin \frac { \pi } { 3 } - 6 \tan \frac { \pi } { 6 } = \\)
(simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression)
b. the answer is undefined

Explanation:

Step1: Recall the values of trigonometric functions

We know that \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\) and \(\tan\frac{\pi}{6}=\frac{\sqrt{3}}{3}\).

Step2: Substitute the values into the expression

Substitute \(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\) and \(\tan\frac{\pi}{6}=\frac{\sqrt{3}}{3}\) into \(2\sin\frac{\pi}{3}-6\tan\frac{\pi}{6}\).
We get \(2\times\frac{\sqrt{3}}{2}-6\times\frac{\sqrt{3}}{3}\).

Step3: Simplify the expression

For \(2\times\frac{\sqrt{3}}{2}\), the \(2\) in the numerator and denominator cancels out, giving \(\sqrt{3}\).
For \(6\times\frac{\sqrt{3}}{3}\), \(6\div3 = 2\), so it becomes \(2\sqrt{3}\).
Then the expression is \(\sqrt{3}-2\sqrt{3}\).

Step4: Combine like - terms

\(\sqrt{3}-2\sqrt{3}=(1 - 2)\sqrt{3}=-\sqrt{3}\).

Answer:

\(2\sin\frac{\pi}{3}-6\tan\frac{\pi}{6}=-\sqrt{3}\)