To evaluate the sine, cosine, and tangent of the angle [tex]\(\theta = -\frac{4\pi}{3}\)[/tex], we need to carefully analyze this angle and use the unit circle.
1. Understanding the Angle:
- The angle [tex]\(\theta = -\frac{4\pi}{3}\)[/tex] is negative, which means it is measured in the clockwise direction from the positive x-axis.
- The equivalent positive angle can be found by adding [tex]\(2\pi\)[/tex] (a full rotation) until the angle falls within the interval [tex]\([0, 2\pi]\)[/tex]:
[tex]\[
-\frac{4\pi}{3} + 2\pi = -\frac{4\pi}{3} + \frac{6\pi}{3} = \frac{2\pi}{3}
\][/tex]
Thus, [tex]\(\theta = -\frac{4\pi}{3}\)[/tex] is coterminal with [tex]\(\frac{2\pi}{3}\)[/tex].
2. Reference Angle:
- The reference angle for [tex]\(\frac{2\pi}{3}\)[/tex] is then found by subtracting from [tex]\(\pi\)[/tex]:
[tex]\[
\pi - \frac{2\pi}{3} = \frac{3\pi}{3} - \frac{2\pi}{3} = \frac{\pi}{3}
\][/tex]
Therefore, the reference angle is [tex]\(\frac{\pi}{3}\)[/tex].
3. Evaluating Trigonometric Functions:
- On the unit circle, [tex]\(\frac{2\pi}{3}\)[/tex] lies in the second quadrant, where sine is positive and cosine is negative.
- Sine:
[tex]\[
\sin\left(\frac{2\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
\][/tex]
Therefore, [tex]\(\sin\left(-\frac{4\pi}{3}\right)\)[/tex] is the same, since sine is an odd function:
[tex]\[
\sin\left(-\theta\right) = -\sin\left(\theta\right) \rightarrow \sin\left(-\frac{4\pi}{3}\right) = -\sin\left(\frac{2\pi}{3}\right) = -\frac{\sqrt{3}}{2} \approx 0.8660254037844385
\][/tex]
- Cosine:
[tex]\[
\cos\left(\frac{2\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) = -\frac{1}{2}
\][/tex]
Therefore, [tex]\(\cos\left(-\frac{4\pi}{3}\right)\)[/tex] is the same, since cosine is an even function:
[tex]\[
\cos\left(-\theta\right) = \cos\left(\theta\right) \rightarrow \cos\left(-\frac{4\pi}{3}\right) = \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \approx -0.5000000000000004
\][/tex]
- Tangent:
[tex]\[
\tan\left(\frac{2\pi}{3}\right) = \frac{\sin\left(\frac{2\pi}{3}\right)}{\cos\left(\frac{2\pi}{3}\right)} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3}
\][/tex]
Therefore, [tex]\(\tan\left(-\frac{4\pi}{3}\right)\)[/tex] is the same, since tangent is an odd function:
[tex]\[
\tan\left(-\theta\right) = -\tan\left(\theta\right) \rightarrow \tan\left(-\frac{4\pi}{3}\right) = -\tan\left(\frac{2\pi}{3}\right) = -(-\sqrt{3}) = \sqrt{3} \approx -1.7320508075688754
\][/tex]
Thus, the values are:
[tex]\[
\sin \theta = 0.8660254037844385
\][/tex]
[tex]\[
\cos \theta = -0.5000000000000004
\][/tex]
[tex]\[
\tan \theta = -1.7320508075688754
\][/tex]
