TMUA Trigonometric Identities

TMUA trigonometric identities are not asked for their own sake; they are asked because rewriting an expression with the right identity makes a harder problem trivial. The candidates who do best on these questions are the ones who recognise the rewrite the question is begging for.

• Pythagorean. $\sin^2\theta + \cos^2\theta = 1$, with the two derived forms $1 + \tan^2\theta = \sec^2\theta$ and $1 + \cot^2\theta = \csc^2\theta$. Used to swap one trig function for another inside an equation.
• Double angle. $\sin 2\theta = 2\sin\theta\cos\theta$; $\cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1$. The three forms of $\cos 2\theta$ all matter; pick whichever brings the equation to a single trig function.
• Compound angle. $\sin(A \pm B)$, $\cos(A \pm B)$, $\tan(A \pm B)$. Recognising a compound-angle expression hiding inside a product or quotient is the standard Paper 1 trick at difficulty 4–5.
• The R formula. $a\sin\theta + b\cos\theta = R\sin(\theta + \alpha)$ with $R = \sqrt{a^2 + b^2}$, $\tan\alpha = b/a$. Turns a two-term expression into a single sine, and from there the max value is just $R$.

The move. Before reaching for algebra, count the trig functions in the expression. If there are two, identify the identity that collapses to one. If there’s a product of sines or cosines at different angles, that’s compound-angle territory.