Reference Angle

When an angle is drawn on the coordinate plane with a vertex at the origin, the reference angle is the angle between the terminal side of the angle and the \(x\)-axis. The reference angle is always between \(0\) and \( \frac{\pi}{2} \) radians (or between \(0\) and \(90\) degrees).

In both these diagrams, the blue angle \(y\) is a reference angle of the red angle \(x.\)

Graphs in trigonometry are cyclic, that is, repeating.

For example, a standard sine wave starts at \( 0 ,\) then repeats the same graph at \( 2\pi ,\) \( 4 \pi ,\) \( 6\pi ,\) etc. So if we're discussing the sine of \( 4\pi ,\) it is identical to the sine of 0.

In order to simplify calculations, then, we can repeatedly add or subtract \( 2\pi \) from an angle until it is in the range \( [0, 2\pi) \) and know that the basic sine, cosine, and tangent will be the same.

However, we can still do better than that! Consider that on the unit circle, the sine is the same as the \(y\)-coordinate. That means if we have the sine of an angle in the second quadrant, it will be identical to the sine of an angle in the first quadrant. Reflecting the angle over the \(y\)-axis preserves the sine; this can also be accomplished by subtracting from \( \pi, \) which gets the same value as the reference angle!

The same idea of equivalence through reflection can allow the reference angles to be used as a proxy for the trigonometric functions across the entire unit circle.

If an angle's terminal side on the \(x\)-axis, the reference angle is 0. If an angle's terminal side is on the \(y\)-axis, the reference angle is \( \frac{\pi}{2} \) (that is, \(90^\circ \) ).

Otherwise, to find the reference angle:

• If the angle is not in the usual range of \( [0, 2\pi) \) or \( [0, 360^\circ) \), either add or subtract \( 2\pi \) (or \( 360^\circ \)) as many times as necessary to put the angle within that range.
• Then use this table, assuming an original angle \(x:\)

The angle \( 205^\circ \) is in Quadrant III, so has a reference angle of \( 205^\circ - 180^\circ = 25^\circ .\)

For \( \frac{11\pi}{3} ,\) first subtract \( 2\pi \): \( \frac{11\pi}{3} - 2\pi = \frac{11\pi}{3} - \frac{6\pi}{3} = \frac{5\pi}{3} .\)

The angle is in Quadrant II, so has a reference angle of \( \pi - \frac{5\pi}{3} = \frac{\pi}{3} .\)

All of the information below can be recreated from the facts that 1.) the sine is the \(y\)-coordinate on the unit circle and 2.) the cosine is the \(x\)-coordinate on the unit circle. The graphic below simply indicate where the \(x\) and \(y\) coordinates are positive or negative on the coordinate plane.

Using the chart above, the rules below then apply.

For sines:

• The sine of an angle in quadrant II is the same as its reference angle; that is, \( \sin(\text{original angle}) = \sin(\text{reference angle}) .\)
• The sine of an angle in quadrant III or IV is the same as the opposite of its reference angle; that is, \( \sin(\text{original angle}) = -\sin(\text{reference angle}) .\)

For cosines:

• The sine of an angle in quadrant IV is the same as its reference angle; that is, \( \cos(\text{original angle}) = \cos(\text{reference angle}) .\)
• The sine of an angle in quadrant II or III is the same as the opposite of its reference angle; that is, \( \cos(\text{original angle}) = -\cos(\text{reference angle}) .\)

Then applying the fact the tangent of an angle is just the sine divided by the cosine:

• The tangent of an angle in quadrant III is the same as its reference angle; that is, \( \tan(\text{original angle}) = \tan(\text{reference angle}) .\)
• The tangent of an angle in quadrant II or IV is the same as its reference angle; that is, \( \tan(\text{original angle}) = -\tan(\text{reference angle}) .\)