Trigonometric Ratios of Angles in Radians

Every angle can be measured in degrees or in radians. A radian measures an angle by the length of arc it cuts on a circle of radius \(1\), and a full turn around that circle is \(2\pi\) radians instead of \(360^\circ\). Once an angle is written in radians, you find its sine, cosine, and tangent exactly the same way you always have: by looking at where the angle lands on the unit circle. Nothing about the ratio itself changes, only the way the angle is labelled.

If you are still getting comfortable moving between the two units, it helps to review how to convert between degrees and radians before working through the examples below, since many problems mix the two.

Place an angle \(\theta\) with its vertex at the origin, measured counterclockwise from the positive \(x\)-axis, and let it meet the unit circle at point \((x, y)\). Then:

\(\sin(\theta) = y\), \(\cos(\theta) = x\), and \(\tan(\theta) = \dfrac{y}{x} = \dfrac{\sin(\theta)}{\cos(\theta)}\).

The most useful radian angles to memorize are the ones built from \(\dfrac{\pi}{6}\), \(\dfrac{\pi}{4}\), and \(\dfrac{\pi}{3}\), because they come straight from 30-60-90 and 45-45-90 triangles. If those triangle ratios are new to you, it is worth reviewing SOH CAH TOA questions first.

Angle (radians)	\(\sin(\theta)\)	\(\cos(\theta)\)	\(\tan(\theta)\)
\(0\)	\(0\)	\(1\)	\(0\)
\(\dfrac{\pi}{6}\)	\(\dfrac{1}{2}\)	\(\dfrac{\sqrt{3}}{2}\)	\(\dfrac{\sqrt{3}}{3}\)
\(\dfrac{\pi}{4}\)	\(\dfrac{\sqrt{2}}{2}\)	\(\dfrac{\sqrt{2}}{2}\)	\(1\)
\(\dfrac{\pi}{3}\)	\(\dfrac{\sqrt{3}}{2}\)	\(\dfrac{1}{2}\)	\(\sqrt{3}\)
\(\dfrac{\pi}{2}\)	\(1\)	\(0\)	undefined
\(\pi\)	\(0\)	\(-1\)	\(0\)
\(\dfrac{3\pi}{2}\)	\(-1\)	\(0\)	undefined
\(2\pi\)	\(0\)	\(1\)	\(0\)

Angles like \(\dfrac{2\pi}{3}\), \(\dfrac{5\pi}{4}\), or \(\dfrac{7\pi}{6}\) do not land in the first quadrant, so you cannot read their ratios straight off a 30-60-90 triangle. Instead, find the reference angle, the acute angle between the terminal side and the \(x\)-axis, and evaluate the ratio for that smaller angle. Then use the ASTC rule to decide whether the final answer should be positive or negative, based on which quadrant the original angle falls in.

Example 1: Evaluate \(\sin\left(\dfrac{2\pi}{3}\right)\)

\(\dfrac{2\pi}{3}\) is between \(\dfrac{\pi}{2}\) and \(\pi\), so it lies in the second quadrant, where sine is positive. Its reference angle is \(\pi - \dfrac{2\pi}{3} = \dfrac{\pi}{3}\). Since \(\sin\left(\dfrac{\pi}{3}\right) = \dfrac{\sqrt{3}}{2}\), and sine stays positive in quadrant two, \(\sin\left(\dfrac{2\pi}{3}\right) = \dfrac{\sqrt{3}}{2}\).

Example 2: Evaluate \(\cos\left(\dfrac{5\pi}{4}\right)\)

\(\dfrac{5\pi}{4}\) is between \(\pi\) and \(\dfrac{3\pi}{2}\), so it is in the third quadrant, where cosine is negative. Its reference angle is \(\dfrac{5\pi}{4} - \pi = \dfrac{\pi}{4}\). Since \(\cos\left(\dfrac{\pi}{4}\right) = \dfrac{\sqrt{2}}{2}\), the answer becomes \(\cos\left(\dfrac{5\pi}{4}\right) = -\dfrac{\sqrt{2}}{2}\).

Example 3: Evaluate \(\tan\left(\dfrac{7\pi}{6}\right)\)

\(\dfrac{7\pi}{6}\) is between \(\pi\) and \(\dfrac{3\pi}{2}\), placing it in the third quadrant, where tangent is positive. The reference angle is \(\dfrac{7\pi}{6} - \pi = \dfrac{\pi}{6}\), and \(\tan\left(\dfrac{\pi}{6}\right) = \dfrac{\sqrt{3}}{3}\). Because tangent is positive in quadrant three, \(\tan\left(\dfrac{7\pi}{6}\right) = \dfrac{\sqrt{3}}{3}\).

Plotting \(y = \sin(x)\) with \(x\) measured in radians shows the repeating wave pattern that trig ratios follow as an angle sweeps around the circle. Notice how the graph touches \(1\) at \(x = \dfrac{\pi}{2}\), crosses zero at \(x = \pi\), reaches \(-1\) at \(x = \dfrac{3\pi}{2}\), and returns to zero after one full period of \(2\pi\).

Cosine and tangent follow their own patterns built from the same unit circle values, but they all share this idea: once you know the ratio for the reference angle and the sign from the quadrant, you can evaluate the ratio for any angle written in radians.

Forgetting to check whether your calculator is set to radian mode is the single most common error when angles are given in radians. A second common slip is mixing up the reference angle formula for each quadrant, so always sketch a quick unit circle picture and confirm which quadrant the angle actually lands in before applying the ASTC rule.

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