From a Known Angle to Its Half
The unit circle gives exact sine and cosine values for familiar angles such as \(30^\circ\), \(45^\circ\), and \(60^\circ\). The half-angle identities let us use those known values to find exact values at angles halfway between them. For instance, halving \(30^\circ\) gives \(15^\circ\), and halving \(45^\circ\) gives \(22.5^\circ\).
The calculation has two important parts. First, use the cosine of the original angle to find the magnitude of the sine or cosine of its half. Then use the quadrant of that half-angle to choose the sign. The radical formulas alone do not determine whether the answer is positive or negative.
The Half-Angle Formulas and Their Signs
The half-angle identities, derived from the double-angle identities, state that
The sign is not chosen by the square root itself: the square root symbol gives a nonnegative value. Instead, choose the sign that agrees with the quadrant of \(\theta/2\). Sine is positive in Quadrants I and II and negative in Quadrants III and IV. Cosine is positive in Quadrants I and IV and negative in Quadrants II and III.
Once sine and cosine are known, tangent can be found from the quotient identity \(\tan x=\sin x/\cos x\), provided \(\cos x\ne0\). Equivalently, when \(1+\cos\theta\ne0\), one may use
Here too, the sign must agree with the quadrant. The quotient form is often a useful check on the tangent value. If \(\cos(\theta/2)=0\), tangent is undefined; in the displayed square-root formula this corresponds to \(1+\cos\theta=0\).
| Quadrant of \(\theta/2\) | Sign of sine | Sign of cosine | Sign of tangent |
|---|---|---|---|
| I | Positive | Positive | Positive |
| II | Positive | Negative | Negative |
| III | Negative | Negative | Positive |
| IV | Negative | Positive | Negative |
To see why the method works, set \(x=\theta/2\), so that \(2x=\theta\). The half-angle identities give the squares of \(\sin x\) and \(\cos x\) in terms of \(\cos(2x)\). Taking square roots gives their magnitudes; the quadrant of \(x\) supplies the signs that squaring has erased. Thus the known value of \(\cos\theta\) determines the magnitudes, and the location of \(\theta/2\) determines the signed values.
Worked Examples
Worked Example: Exact Values at \(15^\circ\)
Find \(\sin15^\circ\), \(\cos15^\circ\), and \(\tan15^\circ\). Since \(15^\circ=30^\circ/2\), use \(\theta=30^\circ\) and \(\cos30^\circ=\sqrt{3}/2\). The angle \(15^\circ\) is in Quadrant I, so both sine and cosine are positive.
For sine,
To express this in the customary form, note that \((\sqrt{6}-\sqrt{2})^2=6+2-2\sqrt{12}=8-4\sqrt{3}=4(2-\sqrt{3})\). Both \(\sqrt{6}-\sqrt{2}\) and \(2\sqrt{2-\sqrt{3}}\) are positive, so they are equal. Therefore,
For cosine, again use the positive sign because \(15^\circ\) is in Quadrant I:
The last equality follows because \((\sqrt{6}+\sqrt{2})^2=8+4\sqrt{3}=4(2+\sqrt{3})\), with both sides positive. Now use the quotient identity:
Check by doubling the angle. The sine double-angle calculation gives \(2\sin15^\circ\cos15^\circ=2(6-2)/16=1/2=\sin30^\circ\). The cosine double-angle calculation gives \(\cos^2 15^\circ-\sin^2 15^\circ=((2+\sqrt{3})-(2-\sqrt{3}))/4=\sqrt{3}/2=\cos30^\circ\). The values are consistent.
Worked Example: Exact Values at \(22.5^\circ\)
Find \(\sin22.5^\circ\), \(\cos22.5^\circ\), and \(\tan22.5^\circ\). Since \(22.5^\circ=45^\circ/2\), use \(\cos45^\circ=\sqrt{2}/2\). The half-angle is in Quadrant I, so sine and cosine are positive.
Use their quotient for tangent. Rationalizing the denominator inside the radical gives
The final equality holds because \((\sqrt{2}-1)^2=3-2\sqrt{2}\), and \(\sqrt{2}-1\) is positive. To check, doubling \(22.5^\circ\) gives \(45^\circ\):
Worked Example: Exact Values at \(165^\circ\)
Find \(\sin165^\circ\), \(\cos165^\circ\), and \(\tan165^\circ\). Because \(165^\circ=330^\circ/2\), use the standard value \(\cos330^\circ=\sqrt{3}/2\). The angle \(165^\circ\) lies in Quadrant II: sine is positive, while cosine and tangent are negative.
The radical simplifications follow from the same squared forms used for \(15^\circ\); the negative sign on cosine is required by Quadrant II. Taking the quotient,
Check by doubling. We obtain \(2\sin165^\circ\cos165^\circ=2(-4)/16=-1/2=\sin330^\circ\), and \(\cos^2 165^\circ-\sin^2 165^\circ=((2+\sqrt{3})-(2-\sqrt{3}))/4=\sqrt{3}/2=\cos330^\circ\). The signs and values agree with the double-angle checks.
A Reliable Calculation Process
These examples use the same sequence of decisions. Keeping the sign decision separate from the radical calculation helps prevent errors.
Write the target as \(\theta/2\), where \(\theta\) is a standard angle whose cosine is known.
Determine its quadrant. This sets the signs of sine, cosine, and tangent.
Substitute the known value of \(\cos\theta\) into the sine and cosine half-angle formulas.
Simplify the radicals, calculate tangent from the quotient if needed, and verify sine and cosine by doubling the angle.
Common Pitfall: Choosing the Positive Root Automatically
A frequent mistake is to treat the square-root symbol as if it settles the sign of the trigonometric value. For example, applying the cosine formula to \(165^\circ=330^\circ/2\) and writing
is incorrect. The radical gives the magnitude, but \(165^\circ\) is in Quadrant II, where cosine is negative. The correction is
The sign can also be checked against the angle itself: \(165^\circ\) is between \(90^\circ\) and \(180^\circ\), so its cosine must be negative. Always locate the half-angle before selecting a sign.
Why These Exact Values Are Useful
Half-angle identities extend the familiar collection of exact trigonometric values. They produce exact values at angles that may not appear in the original standard-angle list, without relying on a calculator approximation. The same method works in any quadrant as long as the starting cosine is known and the quadrant signs are chosen correctly.
A complete answer should include the requested trigonometric value in exact form and a clear sign choice. When more than one value is requested, use the quotient identity to connect tangent to sine and cosine. Checking the sine and cosine by doubling the angle is especially useful: it returns to the known starting angle and tests both the radical calculation and the signs.
Check Your Understanding
Use the half-angle identities, and make the sign decision from the quadrant of the target angle.
- Find \(\sin30^\circ\) by halving \(60^\circ\), and verify it by doubling the angle.
- Find \(\cos75^\circ\) by halving \(150^\circ\). Which sign should the cosine have?
- Find \(\sin120^\circ\) by halving \(240^\circ\). Explain the sign using the quadrant.
- Why is the positive square root not automatically the value of \(\cos165^\circ\)?
- State when the tangent quotient \(\sin(\theta/2)/\cos(\theta/2)\) is undefined.
