The Reference Angle
Completion requirements
The Reference Angle
In the Investigation, you determined the measures of angles formed between the terminal arms of angles and the \(x\)-axis and \(y\)-axis. The angles in standard position follow patterns such that each angle between terminal arm and \(x\)-axis measures \(50^\circ\), and each angle between the terminal arm and the \(y\)-axis measures \(40^\circ\). The patterns produced by the four angles in standard position, \(50^\circ\), \(130^\circ\), \(230^\circ\), and \(310^\circ\), suggest properties that can be used in trigonometry.
The angle formed between the terminal arm of an angle in standard position and the \(x\)-axis has a special name called the reference angle. Note that the reference angle is only formed with the \(x\)-axis, never the \(y\)-axis! This angle is always an acute angle (less than \(90^\circ\)), and will be the focus of determining trigonometric ratios of any angle in Lesson 4.2.
Determining the measure of the reference angle in each quadrant involves calculating the angle between the terminal arm of the angle in standard position and the \(x\)-axis.
Quadrant I:
For an angle terminating in Quadrant I, the reference angle is equal to the angle in standard position.
Quadrant II:
For an angle terminating in Quadrant II, the reference angle is calculated by subtracting the measure of the angle in standard position from \(180^\circ\).
Quadrant III:
For an angle terminating in Quadrant III, the reference angle is calculated by subtracting \(180^\circ\) from the measure of the angle in standard position.
Quadrant IV:
For an angle terminating in Quadrant IV, the reference angle is calculated by subtracting the measure of the angle in standard position from \(360^\circ\).