TMUA Trigonometric Equations and Identities

Identities and exact values

• $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ and $\sin^2\theta+\cos^2\theta=1$.
• Know $\sin$, $\cos$ and $\tan$ of $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$ and $90^\circ$ exactly.

Finding every solution

Find the principal value, then use the symmetry of the graph for the others: $\sin\theta=\sin(180^\circ-\theta)$, $\cos\theta=\cos(360^\circ-\theta)$, and add $180^\circ$ for tangent.

Multiple angles

For $\sin 2x$, first solve for $2x$ over a doubled interval, then halve. Count the solutions before you halve.

Graph shapes and symmetries

Sine and cosine repeat every $360^\circ$ ($2\pi$) and tangent every $180^\circ$ ($\pi$). Also $\sin(-\theta)=-\sin\theta$ and $\cos(-\theta)=\cos\theta$. A quick sketch tells you how many solutions to expect.

If an equation is a quadratic in $\sin x$ or $\cos x$, factorise first. A value outside $-1$ to $1$ gives no solutions. Never cancel a factor of $\sin x$ or $\cos x$ without checking it could be zero.

Always read the required interval carefully. Solutions in $0\le x\le2\pi$ are not the same as solutions in $-\pi\le x\le\pi$, and the end values are included only when the inequality says so.

Worked examples

Common mistakes

• Giving only the principal value and missing the other solutions in the interval.
• Halving the interval for $\sin2x$ instead of doubling it.
• Dividing both sides by $\cos x$ and losing the solutions where $\cos x=0$.
• Mixing degrees and radians.

Test yourself on this topic with our free practice questions.