Question

name:

1. which graph represents an angle in standard position with a measure of 285°?

a.
c.
b.
d.

Explanation:

Step1: Recall standard position angle

An angle in standard position has its vertex at the origin and initial side along the positive x - axis. We measure the angle by rotating the terminal side from the initial side. Angles are positive when rotated counter - clockwise and negative when rotated clockwise.

Step2: Analyze the measure of \(285^{\circ}\)

We know that:

• \(0^{\circ}<\theta < 90^{\circ}\): First quadrant.
• \(90^{\circ}<\theta < 180^{\circ}\): Second quadrant.
• \(180^{\circ}<\theta < 270^{\circ}\): Third quadrant.
• \(270^{\circ}<\theta < 360^{\circ}\): Fourth quadrant.

For \(\theta = 285^{\circ}\), since \(270^{\circ}<285^{\circ}<360^{\circ}\), the terminal side of the angle lies in the fourth quadrant. Also, the angle is measured counter - clockwise from the positive x - axis.
Now let's analyze the options:

• Option A: The angle seems to be in the third quadrant (or has a non - standard orientation), so it is not \(285^{\circ}\).
• Option B: The angle is in the first quadrant (since both x and y coordinates of the terminal side are positive), so it is not \(285^{\circ}\).
• Option C: The angle is measured clockwise (negative angle direction) or has a wrong quadrant orientation.
• Option D: The angle has its initial side on the positive x - axis, is measured counter - clockwise, and the terminal side is in the fourth quadrant (since \(270^{\circ}<285^{\circ}<360^{\circ}\)), which matches the requirements for a \(285^{\circ}\) angle in standard position.

Answer:

D