Reference Angle

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Reference Angle: meaning, method and solved example (Algebra 2) – Maths PDF

In short: A reference angle is the small acute angle a terminal side makes with the horizontal axis, a handy trick for borrowing first-quadrant trig values in any other quadrant.

Algebra 2TrigonometryF-TF.A.2

What is Reference Angle?

A reference angle is a shortcut back to the first quadrant. No matter how large or how negative an angle is, its terminal side makes a small acute angle with the x-axis, and the trig values there match the original ones up to a sign.

Its size stays between \(0^{\circ}\) and \(90^{\circ}\) (or \(0\) and \(\frac{\pi}{2}\)). For \(\theta\) in the window \(0^{\circ}\) to \(360^{\circ}\), the formulas are: \(\theta\) in Quadrant I, \(180^{\circ} – \theta\) in II, \(\theta – 180^{\circ}\) in III and \(360^{\circ} – \theta\) in IV.

How to spot it

  • An exact trig value is requested for an angle like 207° or 5π/6 that is outside the first quadrant.
  • A problem wants the small positive gap between where an angle ends and the horizontal axis.
  • A negative angle or an angle above 360° must be handled before its sine or cosine is found.
  • You know the value of a trig function in Quadrant I and need the matching value in another quadrant.

The method

  1. Bring the angle into one turn by adding or subtracting 360° (or 2π) until it sits between 0° and 360°.
  2. Name the quadrant in which the angle finishes.
  3. Use that quadrant’s formula to get the acute gap to the x-axis.
  4. Compute the trig value of that acute angle, then attach the sign the original quadrant requires.
negative x-axisterminal side27°angle = 207°reference angle = 207° – 180° = 27°
For 207° in Quadrant III, the reference angle is 207° – 180° = 27°.

Reference Angle: solved example

Problem

Find the reference angle for \(312^{\circ}\) and use it to evaluate \(\sin 312^{\circ}\) to four decimal places.

Solution

An angle of \(312^{\circ}\) ends in Quadrant IV, so the reference angle is \(360^{\circ} – 312^{\circ} = 48^{\circ}\).

Sine is negative in Quadrant IV, and \(\sin 48^{\circ} \approx 0.7431\). Therefore the reference angle is \(48^{\circ}\) and \(\sin 312^{\circ} \approx -0.7431\).

Try it

Question 1

Find the reference angle for \(\frac{5\pi}{4}\).

Show the answer

It lies in Quadrant III, so subtract \(\pi\): \(\frac{5\pi}{4} – \pi = \frac{\pi}{4}\).

Question 2

Find the reference angle for \(-110^{\circ}\).

Show the answer

A coterminal angle is \(250^{\circ}\), in Quadrant III, so \(250^{\circ} – 180^{\circ} = 70^{\circ}\).

Reference Angle vs. Coterminal Angles

Coterminal angles share the very same terminal side and differ by full turns, so they can be large. A reference angle is a single small acute angle measured to the x-axis. Angles 207° and -153° are coterminal, and both have the same reference angle, 27°.

Coterminal Angles

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