Deriving the Half-Angle Identities

Why Half-Angle Formulas Need a Sign

The cosine double-angle identity relates the cosine of an angle to the sine and cosine of half that angle. Rearranging it gives formulas for the squares of those half-angle values. To obtain the values themselves, we take square roots—but a square root alone does not tell us whether the original value was positive or negative.

That distinction is essential. The principal square root is always nonnegative, while sine and cosine can be positive, negative, or zero depending on the angle. The half-angle identities therefore have two parts: an algebraic formula for the square, and a sign choice determined by the quadrant of the half-angle.

Deriving the Squared Identities

The cosine double-angle identity has two useful forms:

To isolate \(\sin^2 x\), begin with the first form. Add \(2\sin^2 x\) to both sides and subtract \(\cos(2x)\):

To isolate \(\cos^2 x\), use the second form. Add \(1\) to both sides and divide by \(2\):

Now set \(x=\theta/2\), so \(2x=\theta\). This gives the squared half-angle identities.

The identities hold for every real \(\theta\), because they follow by algebraic rearrangement of the double-angle identity, which itself holds for every real angle. The right-hand sides are nonnegative: since \(-1\leq \cos\theta\leq 1\), both \(1-\cos\theta\) and \(1+\cos\theta\) are at least zero.

From Squares to Signed Values

Taking square roots of the squared identities gives the possible magnitudes of the half-angle sine and cosine. The square-root symbol \(\sqrt{\phantom{x}}\) means the nonnegative principal square root. Consequently, the signed formulas need a separate sign choice:

The sign is not chosen by the value of \(\cos\theta\) alone. It is chosen by the signs of sine and cosine at \(\theta/2\). In particular, locate the half-angle in a quadrant, using any information given about \(\theta\). If the angle is outside a standard \(0\) to \(2\pi\) cycle, reduce \(\theta/2\) by a whole-number multiple of \(2\pi\) to identify its position on the unit circle. Such a reduction does not change its sine or cosine.

If \(\theta/2\) lies on an axis rather than in a quadrant, one or both values may be zero. The squared identities still apply; the sign choice is then unambiguous because the corresponding square root is zero.

A Complete Justification for the Sign Rule

The squared identities determine the absolute values of the half-angle sine and cosine. For any real number \(a\), the nonnegative square root of \(a^2\) is \(|a|\), not necessarily \(a\). Thus:

The quadrant tells us whether each original value equals its absolute value or the negative of its absolute value. This gives the following procedure.

Worked Examples

Common Pitfall: Treating the Square Root as the Answer

A frequent error is to take the positive square root and assume it is automatically the value of the trigonometric function. For example, in the second worked example, the cosine identity gives:

The calculation \(\cos(\theta/2)=1/\sqrt{5}\) is wrong for that example: the half-angle is in Quadrant II, where cosine is negative. The correction is to use the principal square root only for the magnitude, then apply the quadrant sign:

A second related mistake is to try to decide the half-angle sign from \(\cos\theta\) alone. The given cosine helps evaluate the squared identity, but the interval or other angle information is what locates \(\theta/2\) and determines its signs.

Why These Identities Are Useful

The identities convert a trigonometric value at \(\theta\) into the squares of sine and cosine at \(\theta/2\). That makes them useful whenever a problem supplies information about \(\cos\theta\) but asks about half the angle. The formulas also illustrate a general point in algebra: an equation for a square determines a magnitude, while additional information is needed to recover the sign.

Keep the two tasks separate. First calculate the squared value accurately. Then use the location of the half-angle to choose the sign. If either task is skipped, a numerically plausible square root may still be the wrong trigonometric value.