Reference and Coterminal Angles

What are coterminal angles? These are two or more angles in standard position with the same terminal side. Standard position is where the vertex is at the origin and the initial side along the positive x-axis. These angles are found by taking the measure of the angle, and adding multiples of 360 degrees or 2π. Every angle has infinite coterminal angles.

Coterminal angles example Image source: Coterminal Angles

What is a reference angle? This is the acute angle formed by the terminal side of the given angle and the X-axis. This is only if the angle is not one of the four quadrantal angles (90°, 180°, 270°, 360°). If the angle is greater than 360°, or less than 0°, the reference angle can be found by using a coterminal angle.

Reference angle example Image source: Reference Angles

Reference Angle Formulas Quadrant 1 = Actual Quadrant 2 = 180° – Angle in radians = π – Angle Quadrant 3 = Angle – 180° in radians = Angle – π Quadrant 4 = 360° – angle in radians = 2π – angle

Example 1 Q: If the angle measure is 50° in standard position, find one coterminal angle. A: 50 + 360 = 410°

Example 2 Q: If the angle measure is 110° in standard position, find the reference angle. A: 180 – 110 = 70°

Example 3 Q: If the angle measure is –120°, find the one coterminal angle and reference. A: –120 + 360 = 240°(coterminal) 240 – 180 = 60° (reference)

Example 4 Q: If the angle measure is 279°, find two coterminal angles and the reference angle. A: 279 + 360 = 639° (coterminal) 279 + (360(2)) = 999°(coterminal) 360 – 279 = 81° (reference)

Extra Examples: 1. 500° 500 – 360 = 140° 140 – 360 = –220° Determine a positive and negative coterminal angle. 1. 500°   500 – 360 = 140° 140 – 360 = –220°

Extra Examples: 2. 7𝜋 5 7𝜋 5 + 2𝜋 = 7𝜋 5 + 10𝜋 5 = 17𝜋 5 7𝜋 5 – 2𝜋 = 2. 7𝜋 5 7𝜋 5 + 2𝜋 = 7𝜋 5 + 10𝜋 5 = 17𝜋 5 7𝜋 5 – 2𝜋 = 7𝜋 5 – 10𝜋 5 = – 3𝜋 5

Extra Examples: Find the reference angle. 60° 400° 5. 160° 60° 400 – 360 = 40° Quadrant: II 180 – 160 = 20°