Find Reference Angle - Free Mathematics Tutorials

Find the Reference Angle

The reference angle of an angle in standard position is the acute angle formed between the terminal side of the angle and the x-axis. Two or more coterminal angles share the same reference angle.

Assume angle \(A\) is positive and less than \( 360^\circ \) (or \(2\pi\) radians). The reference angle \(A_r\) depends on the quadrant:

• Quadrant I: \[A_r = A\]
• Quadrant II: \[ A_r = 180^\circ - A \quad \text{(degrees)}, \quad A_r = \pi - A \quad \text{(radians)} \]
• Quadrant III: \[ A_r = A - 180^\circ \quad \text{(degrees)}, \quad A_r = A - \pi \quad \text{(radians)} \]
• Quadrant IV: \[ A_r = 360^\circ - A \quad \text{(degrees)}, \quad A_r = 2\pi - A \quad \text{(radians)} \]

Examples

Example 1: Find the reference angle for \(A = 120^\circ\).

Solution: Angle \(A\) is in quadrant II. Using the formula for quadrant II:

\[ A_r = 180^\circ - 120^\circ = 60^\circ \]

The reference angle is \(60^\circ\).

Example 2: Find the reference angle for \(A = -\frac{15\pi}{4}\).

Solution: The angle is negative. Find a coterminal angle between 0 and \(2\pi\):

\[ A_c = -\frac{15\pi}{4} + 2(2\pi) = \frac{\pi}{4} \]

Since \(A\) and \(A_c\) are coterminal, they share the same reference angle. \(A_c\) is in quadrant I:

\[ A_r = A_c = \frac{\pi}{4} \]

Example 3: Find the reference angle for \(A = -30^\circ\).

Solution: Angle \(A\) is negative and in quadrant IV. The reference angle is the absolute value:

\[ A_r = |-30^\circ| = 30^\circ \]

Exercises

Find the reference angles for:

1. \(A = 1620^\circ\)
2. \(A = -\frac{29\pi}{6}\)
3. \(A = -\frac{\pi}{7}\)

Solutions:

1. \(A_r = 25^\circ\)
2. \(A_r = \frac{\pi}{6}\)
3. \(A_r = \frac{\pi}{7}\)

Related Resources

• Step-by-Step Solver to Find Coterminal Angles
• Step-by-Step Solver to Find Reference Angles
• Angles in Trigonometry
• Reference Angle Calculator
• Coterminal Angle Calculator
• Quadrant Finder Calculator