Questions: From the information given, find the quadrant in which the terminal point determined III, or IV. (a) sin(t)<0 and cos(t)<0, quadrant ; (b) sin(t)>0 and cos(t)<0, quadrant ; (c) sin(t)>0 and cos(t)>0, quadrant ; (d) sin(t)<0 and cos(t)>0, quadrant ;

From the information given, find the quadrant in which the terminal point determined III, or IV.
(a) sin(t)<0 and cos(t)<0, quadrant ;
(b) sin(t)>0 and cos(t)<0, quadrant ;
(c) sin(t)>0 and cos(t)>0, quadrant ;
(d) sin(t)<0 and cos(t)>0, quadrant ;

Transcript text: From the information given, find the quadrant in which the terminal point determined III, or IV. (a) $\sin (t)<0$ and $\cos (t)<0$, quadrant $\square$ ; (b) $\sin (t)>0$ and $\cos (t)<0$, quadrant $\square$ ; (c) $\sin (t)>0$ and $\cos (t)>0$, quadrant $\square$ ; (d) $\sin (t)<0$ and $\cos (t)>0$, quadrant $\square$ ;

Solution

Solution Steps

Step 1: Determine the sign of $\sin(t)$ and $\cos(t)$

The sign of $\sin(t)$ is given as negative, and the sign of $\cos(t)$ is given as negative.

Step 2: Apply the quadrant determination rule

Based on the signs of $\sin(t)$ and $\cos(t)$, we apply the following rules:

If $\sin(t) > 0$ and $\cos(t) > 0$, the terminal point lies in Quadrant I.
If $\sin(t) > 0$ and $\cos(t) < 0$, the terminal point lies in Quadrant II.
If $\sin(t) < 0$ and $\cos(t) < 0$, the terminal point lies in Quadrant III.
If $\sin(t) < 0$ and $\cos(t) > 0$, the terminal point lies in Quadrant IV.

Final Answer:

Based on the given signs, the terminal point determined by angle $t$ lies in Quadrant III.

Step 1: Determine the sign of $\sin(t)$ and $\cos(t)$

The sign of $\sin(t)$ is given as positive, and the sign of $\cos(t)$ is given as negative.

Step 2: Apply the quadrant determination rule

Based on the signs of $\sin(t)$ and $\cos(t)$, we apply the following rules:

If $\sin(t) > 0$ and $\cos(t) > 0$, the terminal point lies in Quadrant I.
If $\sin(t) > 0$ and $\cos(t) < 0$, the terminal point lies in Quadrant II.
If $\sin(t) < 0$ and $\cos(t) < 0$, the terminal point lies in Quadrant III.
If $\sin(t) < 0$ and $\cos(t) > 0$, the terminal point lies in Quadrant IV.

Final Answer:

Based on the given signs, the terminal point determined by angle $t$ lies in Quadrant II.

Step 1: Determine the sign of $\sin(t)$ and $\cos(t)$

The sign of $\sin(t)$ is given as positive, and the sign of $\cos(t)$ is given as positive.

Step 2: Apply the quadrant determination rule

Based on the signs of $\sin(t)$ and $\cos(t)$, we apply the following rules:

If $\sin(t) > 0$ and $\cos(t) > 0$, the terminal point lies in Quadrant I.
If $\sin(t) > 0$ and $\cos(t) < 0$, the terminal point lies in Quadrant II.
If $\sin(t) < 0$ and $\cos(t) < 0$, the terminal point lies in Quadrant III.
If $\sin(t) < 0$ and $\cos(t) > 0$, the terminal point lies in Quadrant IV.

Final Answer:

Based on the given signs, the terminal point determined by angle $t$ lies in Quadrant I.

Step 1: Determine the sign of $\sin(t)$ and $\cos(t)$

The sign of $\sin(t)$ is given as negative, and the sign of $\cos(t)$ is given as positive.

Step 2: Apply the quadrant determination rule

Based on the signs of $\sin(t)$ and $\cos(t)$, we apply the following rules:

If $\sin(t) > 0$ and $\cos(t) > 0$, the terminal point lies in Quadrant I.
If $\sin(t) > 0$ and $\cos(t) < 0$, the terminal point lies in Quadrant II.
If $\sin(t) < 0$ and $\cos(t) < 0$, the terminal point lies in Quadrant III.
If $\sin(t) < 0$ and $\cos(t) > 0$, the terminal point lies in Quadrant IV.

Final Answer:

Based on the given signs, the terminal point determined by angle $t$ lies in Quadrant IV.

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