Concept Check Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable. ―90 °

Verified step by step guidance

1. Start by sketching the angle of \(-90^\circ\) in standard position. Recall that an angle in standard position has its vertex at the origin, its initial side along the positive x-axis, and the angle is measured by rotating the initial side.
2. Since the angle is \(-90^\circ\), draw an arrow starting from the positive x-axis and rotating clockwise (because the angle is negative) by 90 degrees. This will place the terminal side along the negative y-axis.
3. To find a positive coterminal angle, add \(360^\circ\) to \(-90^\circ\): \(-90^\circ + 360^\circ = 270^\circ\). Sketch this angle by rotating counterclockwise 270 degrees from the positive x-axis. The terminal side will be the same as the original angle.
4. To find a negative coterminal angle, subtract \(360^\circ\) from \(-90^\circ\): \(-90^\circ - 360^\circ = -450^\circ\). This means rotating clockwise 450 degrees from the positive x-axis, which also ends at the same terminal side.
5. Determine the quadrant for each angle: The original angle \(-90^\circ\) and its coterminal angles \(270^\circ\) and \(-450^\circ\) all have their terminal side on the negative y-axis, which lies on the boundary between Quadrants III and IV, so it is not inside any quadrant but on the axis.

Key Concepts

Standard Position of an Angle

An angle is in standard position when its vertex is at the origin of the coordinate plane and its initial side lies along the positive x-axis. The angle is measured by rotating the initial side to the terminal side, either counterclockwise for positive angles or clockwise for negative angles.

Coterminal Angles

Coterminal angles share the same terminal side but differ by full rotations of 360°. To find coterminal angles, add or subtract multiples of 360° from the given angle. For example, angles of -90°, 270°, and 630° are coterminal because they end at the same position.

Quadrants and Angle Location

The coordinate plane is divided into four quadrants, each defined by the signs of x and y coordinates. The quadrant of an angle depends on where its terminal side lies after rotation. For example, a -90° angle ends on the negative y-axis, which is the boundary between Quadrants III and IV.