The problem asks us to determine the quadrant in which the terminal arm of angle $\theta$ lies, given different conditions involving trigonometric functions of $\theta$. We need to consider the signs of sine, cosine, and tangent in each quadrant. The quadrants are numbered counterclockwise starting from the upper right: * Quadrant I: $0^\circ < \theta < 90^\circ$ * Quadrant II: $90^\circ < \theta < 180^\circ$ * Quadrant III: $180^\circ < \theta < 270^\circ$ * Quadrant IV: $270^\circ < \theta < 360^\circ$ We are given the following conditions: (a) $\sin \theta > 0$ and $\cos \theta > 0$ (b) $\sin \theta < 0$ and $\cos \theta < 0$ (c) $\tan \theta > 0$ and $\cos \theta < 0$ (d) $\tan \theta < 0$ and $\cos \theta < 0$ (e) $\sin \theta < 0$ and $\theta \in [90^\circ, 270^\circ]$ (f) $\cos \theta < 0$ and $0^\circ < \theta < 180^\circ$

1. Problem Description

The problem asks us to determine the quadrant in which the terminal arm of angle $\theta$ lies, given different conditions involving trigonometric functions of $\theta$. We need to consider the signs of sine, cosine, and tangent in each quadrant. The quadrants are numbered counterclockwise starting from the upper right:

* Quadrant I: $0^\circ < \theta < 90^\circ$

* Quadrant II: $90^\circ < \theta < 180^\circ$

* Quadrant III: $180^\circ < \theta < 270^\circ$

* Quadrant IV: $270^\circ < \theta < 360^\circ$

We are given the following conditions:

(a) $\sin \theta > 0$ and $\cos \theta > 0$

(b) $\sin \theta < 0$ and $\cos \theta < 0$

(c) $\tan \theta > 0$ and $\cos \theta < 0$

(d) $\tan \theta < 0$ and $\cos \theta < 0$

(e) $\sin \theta < 0$ and $\theta \in [90^\circ, 270^\circ]$

(f) $\cos \theta < 0$ and $0^\circ < \theta < 180^\circ$

2. Solution Steps

We will analyze each case:

(a) $\sin \theta > 0$ and $\cos \theta > 0$. Sine is positive in Quadrants I and II. Cosine is positive in Quadrants I and IV. Therefore, both are positive in Quadrant I.

(b) $\sin \theta < 0$ and $\cos \theta < 0$. Sine is negative in Quadrants III and IV. Cosine is negative in Quadrants II and III. Therefore, both are negative in Quadrant III.

(c) $\tan \theta > 0$ and $\cos \theta < 0$. Tangent is positive in Quadrants I and III. Cosine is negative in Quadrants II and III. Therefore, both conditions are satisfied in Quadrant III.

(d) $\tan \theta < 0$ and $\cos \theta < 0$. Tangent is negative in Quadrants II and IV. Cosine is negative in Quadrants II and III. Therefore, both are negative in Quadrant II.

(e) $\sin \theta < 0$ and $\theta \in [90^\circ, 270^\circ]$. Sine is negative in Quadrants III and IV. $\theta \in [90^\circ, 270^\circ]$ means the angle is in Quadrants II or III. The intersection of these conditions is Quadrant III.

(f) $\cos \theta < 0$ and $0^\circ < \theta < 180^\circ$. Cosine is negative in Quadrants II and III. $0^\circ < \theta < 180^\circ$ means the angle is in Quadrants I or II. The intersection of these conditions is Quadrant II.

3. Final Answer

(a) Quadrant I

(b) Quadrant III

(c) Quadrant III

(d) Quadrant II

(e) Quadrant III

(f) Quadrant II