Why Reference Angles Are Useful
A reference angle is a small, positive angle that measures how far a nonquadrantal angle’s terminal side is from the nearest horizontal axis. It lets us describe an angle’s position using an acute angle together with its quadrant.
This is useful because the quadrant tells us the signs of the trigonometric functions, while the reference angle records the terminal side’s acute angle with the horizontal axis. In this tutorial, the goal is to find that angle and those signs. Finding exact trigonometric values at particular angles is a separate task.
The rules apply to angles measured in either degrees or radians. Before using them, we first locate the terminal side within one full rotation.
Quadrants, Coterminal Angles, and Axes
An angle in standard position has its vertex at the origin and its initial side along the positive \(x\)-axis. Its terminal side is where the angle ends after rotation. The coordinate axes divide the plane into four quadrants, numbered counterclockwise from the upper-right region.
Angles that differ by a whole number of full rotations have the same terminal side. A full rotation is \(360^\circ\), or \(2\pi\) radians. For example, \(40^\circ\) and \(400^\circ\) have the same terminal side because \(400^\circ-40^\circ=360^\circ\).
To use the reference-angle formulas, choose a coterminal angle \(\theta_0\) in the interval from \(0^\circ\) up to, but not including, \(360^\circ\), or from \(0\) up to, but not including, \(2\pi\). If \(\theta_0\) is not quadrantal, its quadrant determines the formula.
Reference-Angle Formulas
In the first quadrant, the angle itself is already acute. In the other quadrants, subtract from or compare with the nearest horizontal axis. The formulas below use \(\theta_0\), the coterminal angle in one rotation.
| Quadrant | Degrees: reference angle \(\alpha\) | Radians: reference angle \(\alpha\) |
|---|---|---|
| I | \(\alpha=\theta_0\) | \(\alpha=\theta_0\) |
| II | \(\alpha=180^\circ-\theta_0\) | \(\alpha=\pi-\theta_0\) |
| III | \(\alpha=\theta_0-180^\circ\) | \(\alpha=\theta_0-\pi\) |
| IV | \(\alpha=360^\circ-\theta_0\) | \(\alpha=2\pi-\theta_0\) |
These formulas always produce an acute angle for a nonquadrantal \(\theta_0\). For instance, in quadrant III, \(180^\circ<\theta_0<270^\circ\), so \(0^\circ<\theta_0-180^\circ<90^\circ\). The corresponding radian inequalities are \(\pi<\theta_0<3\pi/2\), giving \(0<\theta_0-\pi<\pi/2\).
Why the Formulas Work
Each formula measures the angle from the terminal side to the nearest horizontal axis. In quadrant II, the terminal side lies between the positive \(y\)-axis and the negative \(x\)-axis. The angle from the positive \(x\)-axis to the negative \(x\)-axis is \(180^\circ\), so the remaining angle to the terminal side is \(180^\circ-\theta_0\). In radians, that straight angle is \(\pi\), giving \(\pi-\theta_0\).
The same geometric comparison gives the other cases. In quadrant III, the angle past the negative \(x\)-axis is \(\theta_0-180^\circ\), or \(\theta_0-\pi\). In quadrant IV, the remaining angle before a full turn is \(360^\circ-\theta_0\), or \(2\pi-\theta_0\). In quadrant I, the angle from the positive \(x\)-axis is already \(\theta_0\). Thus each formula measures the acute separation from the nearest horizontal axis, as required by the definition.
Signs from the Quadrant
On the unit circle, cosine corresponds to the horizontal coordinate and sine to the vertical coordinate. The sign of each coordinate depends on which quadrant contains the terminal side. Tangent is positive when sine and cosine have the same sign and negative when their signs differ; it is undefined on the vertical axes.
| Location | Sine | Cosine | Tangent |
|---|---|---|---|
| Quadrant I | Positive | Positive | Positive |
| Quadrant II | Positive | Negative | Negative |
| Quadrant III | Negative | Negative | Positive |
| Quadrant IV | Negative | Positive | Negative |
The coordinate signs explain the first two columns: in quadrant I, both coordinates are positive; in quadrant II, the vertical coordinate is positive and the horizontal one negative; in quadrant III, both are negative; in quadrant IV, the vertical coordinate is negative and the horizontal one positive. For tangent, the quotient of two quantities with the same sign is positive, while the quotient of quantities with opposite signs is negative.
On the axes, the quadrant sign chart does not apply. On the positive or negative \(x\)-axis, sine is zero and tangent is zero. On the positive or negative \(y\)-axis, cosine is zero and tangent is undefined. These cases are why it is important to check whether an angle is quadrantal before assigning it a quadrant.
A Reliable Method
Keep the angle in degrees or radians throughout the calculation.
Add or subtract full rotations until the angle is between \(0^\circ\) and \(360^\circ\), or between \(0\) and \(2\pi\).
Determine whether the reduced angle lies in a quadrant or on an axis.
Use the formula for the quadrant, then use the sign chart. If the angle is quadrantal, report that it has no acute reference angle under the definition here.
Worked Examples
Worked Example: A Degree Angle in Quadrant II
Find the reference angle and the signs of sine, cosine, and tangent for \(140^\circ\).
The angle is between \(90^\circ\) and \(180^\circ\), so it lies in quadrant II. Its reference angle is therefore
The result is acute, as required. In quadrant II, the vertical coordinate is positive and the horizontal coordinate is negative. Therefore sine is positive and cosine is negative. Their signs differ, so tangent is negative.
Check: The angle is \(40^\circ\) short of \(180^\circ\), since \(140^\circ+40^\circ=180^\circ\). This confirms the reference angle.
Worked Example: A Radian Angle in Quadrant III
Find the reference angle and function signs for \(\frac{5\pi}{4}\).
Since \(\pi=\frac{4\pi}{4}\) and \(\frac{3\pi}{2}=\frac{6\pi}{4}\), the angle satisfies
It lies in quadrant III. The quadrant III formula gives
In quadrant III, both the horizontal and vertical coordinates are negative. Thus cosine and sine are negative, while tangent is positive because the two signs agree.
Check: Adding the reference angle to \(\pi\) recovers the original angle: \(\pi+\frac{\pi}{4}=\frac{4\pi}{4}+\frac{\pi}{4}=\frac{5\pi}{4}\).
Worked Example: A Negative Angle Beyond One Rotation
Find the reference angle and function signs for \(-\frac{9\pi}{4}\). Its magnitude is \(\frac{9\pi}{4}\), which is greater than one full rotation, \(2\pi=\frac{8\pi}{4}\).
Add one full rotation at a time to obtain a coterminal angle in the standard interval:
which is still negative. Add another full rotation:
This lies between \(\frac{3\pi}{2}=\frac{6\pi}{4}\) and \(2\pi=\frac{8\pi}{4}\), so it is in quadrant IV. Its reference angle is
In quadrant IV, sine is negative and cosine is positive, so tangent is negative.
Check: The reduced angle differs from the original by two full rotations: \(\frac{7\pi}{4}-\left(-\frac{9\pi}{4}\right)=\frac{16\pi}{4}=4\pi\). They therefore have the same terminal side.
Worked Example: A Degree Angle in Quadrant IV
Find the reference angle and function signs for \(315^\circ\).
Because \(270^\circ<315^\circ<360^\circ\), the terminal side is in quadrant IV. Use the quadrant IV formula:
The vertical coordinate is negative and the horizontal coordinate is positive in quadrant IV. Thus sine is negative, cosine is positive, and tangent is negative.
Check: The reference angle completes the full turn: \(315^\circ+45^\circ=360^\circ\).
Common Mistake: Using the Wrong Quadrant Formula
A frequent error is to use \(360^\circ-\theta_0\) for every angle outside quadrant I. For \(230^\circ\), this gives \(360^\circ-230^\circ=130^\circ\). But \(130^\circ\) is not acute, and it is not the angle between the terminal side and the nearest horizontal axis.
Since \(180^\circ<230^\circ<270^\circ\), the angle lies in quadrant III. The correct calculation is
Check: The terminal side is \(50^\circ\) past the negative \(x\)-axis, because \(180^\circ+50^\circ=230^\circ\). The incorrect result \(130^\circ\) is not acute, so it cannot be the reference angle.
Why the Distinction Matters
Reference angles and quadrant signs separate two pieces of information that are easy to confuse. The reference angle records the acute geometric angle; the quadrant records which signs apply. Two angles can have the same reference angle while their functions have different signs because their terminal sides lie in different quadrants.
For example, \(45^\circ\) and \(315^\circ\) both have reference angle \(45^\circ\), but they lie in quadrants I and IV. Sine is positive in quadrant I and negative in quadrant IV. A reference angle by itself therefore does not determine the signs; the quadrant must also be known.
When an angle is negative or larger than one full rotation, reduce it first. Then identify the quadrant, select the matching formula, and check that the result is acute. This sequence prevents most reference-angle errors.
Check Your Understanding
For each nonquadrantal angle, find its reference angle and the signs of sine, cosine, and tangent. For any quadrantal angle, state that it lies on an axis and has no acute reference angle under the definition used here.
- Find the reference angle and signs for \(210^\circ\).
- Find the reference angle and signs for \(\frac{11\pi}{6}\).
- Reduce \(-\frac{5\pi}{3}\) to an angle in the standard interval, then find its reference angle and signs.
- Is \(360^\circ\) in a quadrant? Explain.
- For an angle in quadrant II, which of sine, cosine, and tangent are positive?
