Trigonometric identities & equations

Each identity has one standard job:

• $\tan \theta =\sin \theta \mathrm{/}\cos \theta$ converts a mixed $\sin$/$\cos$ equation into one function. Seeing $3\sin \theta =2\cos \theta$ should trigger “divide by $\cos \theta$” → $\tan \theta =\frac{2}{3}$. (Dividing by $\cos \theta$ is safe here because $\cos \theta =0$ doesn’t satisfy the original equation — see the mistakes section for when it isn’t.)
• ${\sin }^2\theta +{\cos }^2\theta =1$ swaps a squared function for the other one, turning equations like $2{\cos }^2\theta +\sin \theta =1$ into a quadratic in $\sin \theta$.

A calculator gives one angle — ${\sin }^{-1},{\cos }^{-1},{\tan }^{-1}$ return the principal value. The range (usually $0^{\circ }\le \theta \le {360}^{\circ }$) holds more. Where each function repeats:

• sin: second solution at ${180}^{\circ }-\theta$
• cos: second solution at ${360}^{\circ }-\theta$
• tan: repeats every ${180}^{\circ }$

A sketch of the curve — or the CAST quadrant diagram, whichever you think in — takes ten seconds and shows exactly how many answers the range holds. Count them before solving; then you know when you’re done.

“Show that” questions want a chain of equalities from one side to the other. The working rules:

1. Start from one side (usually the messier one) and transform it — don’t manipulate both sides at once as if solving an equation.
2. Convert $\tan$ to $\sin \mathrm{/}\cos$ early; combine fractions over a common denominator; deploy ${\sin }^2+{\cos }^2=1$ whenever a $1$, ${\sin }^2$ or ${\cos }^2$ needs to become the others.
3. End by writing the target expression exactly — the final line is the mark.

1. 1.Dividing the equation by sinθ or cosθ

   $\sin \theta \cos \theta =\sin \theta$ divided by $\sin \theta$ loses every solution of $\sin \theta =0$. Move everything to one side and factorise instead: $\sin \theta (\cos \theta -1)=0$ keeps both families.

2. 2.Stopping at the calculator's answer

   The inverse functions return one principal value; the range usually holds two or three solutions. Sketch or CAST first, count the crossings, and only stop when you have that many.

3. 3.Working in degrees when the question is in radians

   If the range reads $0\le \theta \le 2\pi$, answer in radians — exact multiples of $\pi$ where possible. Check the calculator mode matches before the first keystroke.

4. 4.Discarding the 'impossible' root without saying why

   When a quadratic gives $\sin \theta =1.4$, reject it in writing(“no solutions since $-1\le \sin \theta \le 1$”) — the rejection is part of the argument, and silent omission looks like an error.

5. 5.Treating an identity proof like an equation

   You can’t “do the same to both sides” of something you’re trying to prove. Transform one side until it becomes the other; each line must follow from the last.