Math

QuestionDetermine the quadrant where tanθ<0\tan \theta < 0 and sinθ<0\sin \theta < 0.

Studdy Solution

STEP 1

Assumptions1. The angle θ\theta is in standard position (its initial side is on the positive x-axis). . The trigonometric functions sine and tangent are defined as follows in the context of a unit circle sinθ\sin \theta is the y-coordinate of the point where the terminal side of the angle intersects the unit circle, and tanθ=sinθcosθ\tan \theta = \frac{\sin \theta}{\cos \theta}.

STEP 2

We are given that tanθ<0\tan \theta <0. The tangent function is negative in the second and fourth quadrants.

STEP 3

We are also given that sinθ<0\sin \theta <0. The sine function is negative in the third and fourth quadrants.

STEP 4

The only quadrant where both conditions are met (i.e., both tanθ<0\tan \theta <0 and sinθ<0\sin \theta <0) is the fourth quadrant.
So, the angle θ\theta is in the fourth quadrant.

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