QuestionFind the exact value of $\tan^{-1}(-1)$ .

Studdy Solution

STEP 1

Assumptions1. The problem is asking for the exact value of the inverse tangent of -1, denoted as $\tan^{-1}(-1)$ . . We are working in the standard mathematical convention where angles are measured in radians.

STEP 2

We need to find an angle whose tangent is -1. We know that the tangent of an angle in the unit circle is defined as the ratio of the y-coordinate to the x-coordinate.

STEP 3

We know that $\tan(\theta) = -1$ when $\theta$ is an odd multiple of $\frac{\pi}{}$ in the second and fourth quadrants, where the tangent function is negative.

STEP 4

The smallest positive angle that satisfies this condition is $\frac{3\pi}{4}$ in the second quadrant. However, the range of $\tan^{-1}(x)$ is $(-\frac{\pi}{2}, \frac{\pi}{2})$ , so we need to find an angle in this range.

STEP 5

The angle $\theta = -\frac{\pi}{4}$ in the fourth quadrant also has a tangent of -1, and it lies in the range of $\tan^{-1}(x)$ .

STEP 6

Therefore, the exact value of the expression $\tan^{-1}(-1)$ is $-\frac{\pi}{4}$ .
The exact value of the expression $\tan^{-1}(-1)$ is $-\frac{\pi}{4}$ .

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