Double Angle Identities

Double-angle identities rewrite trigonometric functions of 2θ in terms of functions of θ.

Interpret → represent → solve → check. The problem is translated into a mathematical representation, solved with an appropriate operation, and checked against the original quantities and conditions.

Doubling an angle does not double its sine or cosine; it creates specific products and squares.

Sine double-angle identity

Set $A=B=\theta$ in the sine sum identity:

\sin(2\theta) =\sin\theta\cos\theta+\cos\theta\sin\theta =2\sin\theta\cos\theta.

Thus

\boxed{\sin2\theta=2\sin\theta\cos\theta}.

Worked example from a triangle ratio

The signs match because $2\theta$ is acute for the triangle represented here.

Choosing a cosine form

If only $\sin\theta$ is known, $1-2\sin^2\theta$ avoids finding cosine. If only cosine is known, use $2\cos^2\theta-1$. If both are known, $\cos^2\theta-\sin^2\theta$ may be direct.

All three are identical, so differing results reveal an arithmetic or sign error.

Solving equations

Double-angle identities can convert an equation to one trig function. For example,

\cos2x=1-2\sin^2x

may turn an equation involving both $\cos2x$ and $\sin x$ into a quadratic in $\sin x$. After solving, recover every angle in the stated interval.

Common mistakes

Writing $\sin2\theta=2\sin\theta$. The cosine factor is required.

Writing $\cos2\theta=2\cos\theta$. Use one of the three squared forms.

Assuming $2\theta$ stays in the same quadrant. Double the angle range.

Using a cosine form that introduces an unknown unnecessarily. Choose strategically.

Forgetting domain restrictions in tangent. Its denominator and the original tangent must be defined.

Quick self-check

Which double-angle identity matches the known information?
Is the doubled angle's quadrant determined separately?
Are squared fractions and signs handled accurately?
If solving, have all cycles of the compound angle been included?
Can another equivalent cosine form verify the result?