To find the exact value of [tex]\(\tan \left(-\frac{\pi}{3}\right)\)[/tex], we can follow these steps:
1. Understand the angle: The given angle is [tex]\(-\frac{\pi}{3}\)[/tex].
2. Locate the angle on the unit circle: An angle of [tex]\(-\frac{\pi}{3}\)[/tex] is the same as rotating [tex]\(\frac{\pi}{3}\)[/tex] radians in the clockwise direction.
3. Reference angle: The reference angle for [tex]\(-\frac{\pi}{3}\)[/tex] is [tex]\(\frac{\pi}{3}\)[/tex].
4. Tangent values: The tangent of an angle [tex]\(\theta\)[/tex] is given by the ratio of the sine and cosine of [tex]\(\theta\)[/tex]:
[tex]\[
\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}
\][/tex]
5. Standard values:
- For [tex]\(\frac{\pi}{3}\)[/tex]:
[tex]\[
\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
\][/tex]
[tex]\[
\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
\][/tex]
- Therefore:
[tex]\[
\tan\left(\frac{\pi}{3}\right) = \frac{\sin\left(\frac{\pi}{3}\right)}{\cos\left(\frac{\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}
\][/tex]
6. Adjustment for the negative angle:
- Tangent is an odd function, which means [tex]\(\tan(-x) = -\tan(x)\)[/tex].
- Therefore:
[tex]\[
\tan\left(-\frac{\pi}{3}\right) = -\tan\left(\frac{\pi}{3}\right) = -\sqrt{3}
\][/tex]
From these calculations, we conclude that:
[tex]\[
\tan \left(-\frac{\pi}{3}\right) = -\sqrt{3}
\][/tex]
So, the exact value is:
[tex]\[ -\sqrt{3} \][/tex]
