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° in standard position. This means drawing the initial side along the positive x-axis and rotating the terminal side 90° counterclockwise, which will point straight up along the positive y-axis.
2. Draw an arrow from the initial side to the terminal side to represent the 90° rotation in the counterclockwise direction, indicating a positive angle.
3. To find a positive coterminal angle, add 360° to 90°, using the formula \(\theta_{coterminal} = \theta + 360k\), where \(k\) is an integer. For \(k=1\), the positive coterminal angle is \$90° + 360° = 450°$.
4. To find a negative coterminal angle, subtract 360° from 90°, again using \(\theta_{coterminal} = \theta + 360k\). For \(k=-1\), the negative coterminal angle is \$90° - 360° = -270°$.
5. Determine the quadrant for each angle: 90° lies on the positive y-axis (between Quadrant I and II), 450° is equivalent to 90° and also lies on the positive y-axis, and -270° also ends at the same position on the positive y-axis. Since these angles lie exactly on the axis, they are not in any quadrant but on the boundary between Quadrants I and II.

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°, 450°, and -270° are coterminal.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants numbered counterclockwise starting from the upper right. The quadrant of an angle depends on the location of its terminal side: 90° lies on the positive y-axis, between Quadrants I and II, so it is not strictly in any quadrant but on the boundary.