Half-angle identities is one of 5 angle formulas formulas in the trigonometry section of this library, and it is used at high school · ap level.
Why half-angle identities works
Start with cos 2A = 1 − 2 sin²A and just rename the angle: let A be θ/2, so 2A is θ. Solving that line for sin²A gives the first identity, and doing the same to cos 2A = 2 cos²A − 1 gives the second. Nothing new is assumed; the double-angle rule is simply read from right to left.
What each symbol means
$\theta$ is an angle.
Half-angle identities: when it holds
Choose the sign of sine or cosine from the quadrant of $\theta/2$.
When it stops applying
Halving can move you into a different quadrant from the one θ lives in, and then the sign you would guess is wrong. With θ = 300 degrees, sine is negative, yet θ/2 = 150 degrees puts the half angle in quadrant II where sine is the positive 0.5.
Half-angle identities: a worked example
$\sin^2(30^\circ)=(1-\cos60^\circ)/2=1/4$.
The mistake to avoid
What people do: Students take the square root and always write the positive value, since the right side of the equation is positive.
Why it goes wrong: The right side computes the square, which is positive no matter what. The actual sine or cosine of θ/2 can still be negative, and the identity has no way to say so.
Do this instead: Locate θ/2 on the circle before you choose a sign. For θ = 500 degrees, θ/2 is 250 degrees, which sits in quadrant III, so both the sine and the cosine of the half angle are negative.
Half-angle identities: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $\theta$ is an angle.
- Check the conditions before substituting. Choose the sign of sine or cosine from the quadrant of $\theta/2$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most trigonometry slips.
Where this formula fits
- Subject
- Trigonometry formulas — 18 entries in this library
- Topic
- Angle formulas
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where half-angle identities comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Trigonometric Identities — the lesson behind this formula: simplify and prove relationships between trig functions.
- Trigonometry Calculator — check your substitution and the value it produces.
- Study trigonometry — the subject guide that explains the ideas these formulas compress.
- Precalculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 18 trigonometry formulas — the full grouped reference, or the complete formula library.
Questions about half-angle identities
Why is the identity written with a square on the left?
Because that is the form the algebra actually produces, and it is honest about the sign. Squaring hides whether the value was positive or negative, so the identity stops there and leaves that choice to you.
Which cosine do I plug in, the full angle or the half angle?
Always the full angle θ. That is the point of the rule: it trades an angle you cannot evaluate for one you can. To get sin 15 degrees you feed in cos 30 degrees.
How do I find the exact value of sin 22.5 degrees?
Use θ = 45 degrees. Then sin²(22.5°) = (1 − √2/2)/2, so sin 22.5° = √(2 − √2)/2, which is about 0.3827. The half angle sits in quadrant I, so keep the positive root.
Is there a half-angle rule for tangent?
Yes, and the neat versions avoid square roots entirely: tan(θ/2) equals sin θ/(1 + cos θ), and also (1 − cos θ)/sin θ. Because there is no root, there is no sign to choose.