Join the IDNLearn.com community and get your questions answered by experts. Our platform offers reliable and comprehensive answers to help you make informed decisions quickly and easily.

Evaluate the sine, cosine, and tangent of the angle.

[tex]\[ \theta = -\frac{4 \pi}{3} \][/tex]

[tex]\[ \sin \theta = \square \][/tex]

[tex]\[ \cos \theta = \square \][/tex]

[tex]\[ \tan \theta = \square \][/tex]


Sagot :

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]