To determine the exact value of [tex]\(\tan \left(\frac{3 \pi}{4}\right)\)[/tex], follow these steps:
1. Understand the angle [tex]\(\frac{3 \pi}{4}\)[/tex]:
- The angle [tex]\(\frac{3 \pi}{4}\)[/tex] radians is in the second quadrant of the unit circle.
- This is equivalent to 135 degrees.
2. Recall the properties of tangent function:
- In the unit circle, the tangent of an angle is defined as [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex].
- In the second quadrant, the sine function is positive, and the cosine function is negative, making the tangent function negative.
3. Use the reference angle:
- The reference angle for [tex]\(\frac{3 \pi}{4}\)[/tex] is [tex]\(\pi/4\)[/tex] (or 45 degrees), as [tex]\(\frac{3 \pi}{4} = \pi - \pi/4\)[/tex].
4. Know the tangent of the reference angle:
- The tangent of [tex]\(\pi/4\)[/tex] (45 degrees) is 1, i.e., [tex]\(\tan(\pi/4) = 1\)[/tex].
5. Determine the sign in the second quadrant:
- Since the tangent function is negative in the second quadrant, [tex]\(\tan \left(\frac{3 \pi}{4}\right) = -\tan \left(\frac{\pi}{4}\right)\)[/tex].
6. Compute the value:
- Therefore, [tex]\(\tan \left(\frac{3 \pi}{4}\right) = -1\)[/tex].
Given the possible answers:
- [tex]\(-2 \sqrt{2}\)[/tex]
- [tex]\(1\)[/tex]
- [tex]\(-1\)[/tex]
- [tex]\(2 \sqrt{2}\)[/tex]
The exact value of [tex]\(\tan \left(\frac{3 \pi}{4}\right)\)[/tex] is [tex]\(-1\)[/tex].
