QuestionFind the quadrant(s) where $\sin \alpha<0$ and $\sec \alpha<0$ . Answer as $1,2,3$ , or 4, separated by commas.

Studdy Solution

STEP 1

Assumptions1. The angle $\alpha$ is in standard position (its vertex is at the origin and its initial side is along the positive x-axis). . The trigonometric functions sine and secant are defined as follows $\sin \alpha$ is the y-coordinate of the point where the terminal side of the angle intersects the unit circle, and $\sec \alpha$ is the reciprocal of the x-coordinate of that point.
3. The signs of the trigonometric functions vary by quadrant in Quadrant I, all functions are positive; in Quadrant II, only sine is positive; in Quadrant III, only tangent is positive; and in Quadrant IV, only cosine is positive.

STEP 2

We are given that $\sin \alpha<0$ and $\sec \alpha<0$ . This means that the y-coordinate and the reciprocal of the x-coordinate of the point where the terminal side of the angle intersects the unit circle are both negative.

STEP 3

Since $\sin \alpha<0$ , the angle $\alpha$ must be in either Quadrant III or Quadrant IV, where the y-coordinate is negative.

STEP 4

Since $\sec \alpha<0$ , the angle $\alpha$ must be in either Quadrant II or Quadrant III, where the x-coordinate is negative (and thus its reciprocal is also negative).

STEP 5

The only quadrant that satisfies both conditions is Quadrant III.
So, the angles $\alpha$ satisfying the conditions $\sin \alpha<0$ and $\sec \alpha<0$ are in Quadrant3.

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