Math / Trigonometry

QuestionFrom the information given, find the quadrant in which the terminal point determined by $t$ lies. Input I, II, 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 $\square$

Studdy Solution

STEP 1

1. The terminal point determined by $t$ is based on the signs of the sine and cosine functions.
2. The sine function ( $\sin(t)$ ) denotes the y-coordinate of the point on the unit circle.
3. The cosine function ( $\cos(t)$ ) denotes the x-coordinate of the point on the unit circle.
4. The unit circle is divided into four quadrants: - Quadrant I: $\sin(t) > 0$ and $\cos(t) > 0$ - Quadrant II: $\sin(t) > 0$ and $\cos(t) < 0$ - Quadrant III: $\sin(t) < 0$ and $\cos(t) < 0$ - Quadrant IV: $\sin(t) < 0$ and $\cos(t) > 0$

STEP 2

1. Determine the quadrant for the condition $\sin(t) < 0$ and $\cos(t) < 0$ .
2. Determine the quadrant for the condition $\sin(t) > 0$ and $\cos(t) < 0$ .
3. Determine the quadrant for the condition $\sin(t) > 0$ and $\cos(t) > 0$ .
4. Determine the quadrant for the condition $\sin(t) < 0$ and $\cos(t) > 0$ .

STEP 3

Identify the quadrant where both $\sin(t) < 0$ and $\cos(t) < 0$ . - Since $\sin(t)$ (y-coordinate) is negative and $\cos(t)$ (x-coordinate) is negative, the terminal point must be in Quadrant III.

STEP 4

Identify the quadrant where both $\sin(t) > 0$ and $\cos(t) < 0$ . - Since $\sin(t)$ (y-coordinate) is positive and $\cos(t)$ (x-coordinate) is negative, the terminal point must be in Quadrant II.

STEP 5

Identify the quadrant where both $\sin(t) > 0$ and $\cos(t) > 0$ . - Since $\sin(t)$ (y-coordinate) is positive and $\cos(t)$ (x-coordinate) is positive, the terminal point must be in Quadrant I.

STEP 6

Identify the quadrant where both $\sin(t) < 0$ and $\cos(t) > 0$ . - Since $\sin(t)$ (y-coordinate) is negative and $\cos(t)$ (x-coordinate) is positive, the terminal point must be in Quadrant IV.
Summary: (a) $\sin (t)<0$ and $\cos (t)<0$ , quadrant III (b) $\sin (t)>0$ and $\cos (t)<0$ , quadrant II (c) $\sin (t)>0$ and $\cos (t)>0$ , quadrant I (d) $\sin (t)<0$ and $\cos (t)>0$ , quadrant IV

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