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. 174 °

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Start by sketching the angle of 174° in standard position. This means drawing an initial side along the positive x-axis and then rotating counterclockwise by 174° to locate the terminal side of the angle.

Draw an arrow from the initial side to the terminal side to represent the positive rotation of 174°.

To find a positive coterminal angle, add 360° to 174°, giving the angle \(174° + 360° = 534°\). This angle shares the same terminal side as 174° but represents a full rotation plus the original angle.

To find a negative coterminal angle, subtract 360° from 174°, giving the angle \(174° - 360° = -186°\). This angle rotates clockwise from the initial side to the same terminal side as 174°.

Determine the quadrant of each angle by considering their terminal sides: 174° lies in Quadrant II (since it is between 90° and 180°), 534° coterminal with 174° also lies in Quadrant II, and -186° lies in Quadrant III (since it is equivalent to 174° rotated clockwise).

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주요 개념

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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. This helps identify equivalent angles in different rotations.

Quadrants and Angle Location

The coordinate plane is divided into four quadrants. The quadrant of an angle depends on the location of its terminal side: 0° to 90° is Quadrant I, 90° to 180° is Quadrant II, 180° to 270° is Quadrant III, and 270° to 360° is Quadrant IV. Knowing the quadrant helps understand the angle's trigonometric properties.