Question

In Exercises 9 and \(10,\) find all the trigonometric values of \(\theta\) with the given conditions. $$\tan \theta=-1, \quad \sin \theta<0$$

Short answer

The solution to the problem is \( \theta = \frac{7\pi}{4} + 2n\pi \) or \( \theta = 315° + n.360° \) where n is an integer.

Step-by-step solution

Understand the given conditions

We are given two conditions: \( \tan \theta = -1 \) and \( \sin \theta < 0 \). The first implies that for some point on the unit circle, the y-coordinate divided by the x-coordinate is -1. The second implies that the y-coordinate of such point is negative which indicates the direction of the angle.

Identify the Quadrants where the given conditions are met

Knowing that Tangent is negative in Q2 and Q4, and sine is negative in Q3 and Q4 of the unit circle. The meeting point for these conditions which \( \tan \theta = -1 \) and \( \sin \theta < 0 \) would be the 4th Quadrant (Q4).

Find the Trigonometric Value of \( \theta \).

Since we have identified that the conditions meet in the 4th Quadrant (Q4), where \(\tan(\theta) = -1\), this happens at \(\theta = \frac{7\pi}{4}\) or \(\theta = 315°\) in the standard position where \(\theta\) rotates counterclockwise from the positive x-axis. The general solution will be \(\theta = 315° + n.360°\) or \(\theta = \frac{7\pi}{4} + 2n\pi \), where n is an integer.