The sum and difference identities for sine, cosine and tangent, the double-angle and half-angle identities, and using them for exact values and proofs.
A tempting mistake
Is \(\sin(30° + 60°)\) equal to \(\sin 30° + \sin 60°\)? The left side is \(\sin 90° = 1\). The right side is \(\frac{1}{2} + \frac{\sqrt{3}}{2} \approx 1.37\). Trig functions don't distribute over addition.
Sum and difference identities
Watch the signs: in the sine formula the sign matches; in the cosine formula it flips.
Double-angle identities
Set \(B = A\) in the sum identities:
The three versions of \(\cos 2A\) are equivalent thanks to the Pythagorean identity; pick whichever fits the problem.
Half-angle identities
Solving the cosine double-angle identity for the half angle gives:
Practice
Quick check
Which expression equals \(\cos 2A\)?
It's one of the three forms of the cosine double-angle identity.
Quick check
Using the difference identity, \(\sin\left(\pi - x\right)\) simplifies to:
\(\sin\pi\cos x - \cos\pi\sin x = 0 - (-1)\sin x = \sin x\).
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