Trigonometric Identities and Equations

This focus area is small and rarely examined for many marks on its own, but solving trigonometric equations is needed again in Year 12 (for stationary points, intersections and periodic models). Expect a 1-mark multiple-choice question or a 2–3 mark "prove" or "solve" part.

• Reciprocal ratios: $\sec \theta$, $\mathrm{cosec}\theta$ and $\cot \theta$, their exact values and where they are undefined.
• Quotient identities: $\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\cot \theta =\frac{\cos \theta }{\sin \theta }$.
• Complementary angle identities such as $\sin (90^{\circ }-\theta )=\cos \theta$.
• Pythagorean identities and how to use them to simplify expressions and prove identities.
• Trigonometric equations on a restricted domain, in degrees or radians, including multiple angles like $2x$ and equations that reduce to quadratics.

Here $(x,y)$ is a point on a circle of radius $r$, as in the definitions of sin, cos and tan. Each reciprocal ratio is undefined where its denominator is zero:

• $\sec \theta$ is undefined when $\cos \theta =0$, for example at $90^{\circ }$ and $270^{\circ }$.
• $\mathrm{cosec}\theta$ and $\cot \theta$ are undefined when $\sin \theta =0$, for example at $0^{\circ }$ and $180^{\circ }$.

A reciprocal ratio has the same sign as the ratio it comes from, so ASTC still works. For exact values, find the ordinary ratio and flip it: $\sin 60^{\circ }=\frac{\sqrt{3}}{2}$, so $\mathrm{cosec}60^{\circ }=\frac{2}{\sqrt{3}}$.

An identity is true for every value of the variable for which both sides are defined. These are the ones to know:

The other three follow the same pattern: $\cot (90^{\circ }-\theta )=\tan \theta$, $\sec (90^{\circ }-\theta )=\mathrm{cosec}\theta$ and $\mathrm{cosec}(90^{\circ }-\theta )=\sec \theta$, each wherever both sides are defined.

The Pythagorean identities come from $x^2+y^2=r^2$ divided by $r^2$:

Divide the first by ${\cos }^{2}x$ to get the second, and by ${\sin }^{2}x$ to get the third. Rearranged forms are just as useful: $1-{\cos }^{2}x={\sin }^{2}x$ and ${\sec }^{2}x-1={\tan }^{2}x$.

How to prove an identity

1. Start with the more complicated side (usually the one with fractions or reciprocal ratios).
2. Rewrite reciprocal ratios and $\tan$ in terms of $\sin$ and $\cos$.
3. Combine fractions over a common denominator.
4. Use a Pythagorean identity to replace $1-{\cos }^{2}x$, $1-{\sin }^{2}x$ and similar.
5. Simplify until you reach the other side, then write "= RHS".

You may simplify the LHS and the RHS separately until they reach the same expression, but never treat the identity as an equation: don't move terms across the equals sign or do the same operation to both sides, because that assumes what you are trying to prove.

Worked example: proving an identity

Prove that $\sec x-\cos x=\sin x\tan x$.

The identity holds wherever $\cos x\ne 0$.

The method:

1. Rearrange to get a single ratio on its own, such as $\sin x=k$.
2. Find the related angle from the size of $k$ (exact value or calculator).
3. Use the sign of $k$ and ASTC to choose the quadrants.
4. Write every solution in the given domain, in the same units as the domain.

If the equation has a multiple angle such as $2x$, first change the domain. If $0\le x\le 2\pi$, then $0\le 2x\le 4\pi$, so you need solutions for two revolutions before you halve them.

Worked example: a multiple angle

Solve $\cos 2x=-\frac{1}{2}$ for $0\le x\le 2\pi$.

The related angle is $\frac{\pi }{3}$, and cosine is negative in quadrants 2 and 3. With $0\le 2x\le 4\pi$:

There are four solutions, because $\cos 2x$ completes two full cycles on this domain.

Worked example: reducing to a quadratic

Solve $2{\cos }^{2}x+3\sin x-3=0$ for $0\le x\le 2\pi$.

The equation mixes $\sin$ and $\cos$, so use ${\cos }^{2}x=1-{\sin }^{2}x$ to write it in $\sin x$ only.

So $\sin x=\frac{1}{2}$ or $\sin x=1$.

• $\sin x=\frac{1}{2}$: related angle $\frac{\pi }{6}$, quadrants 1 and 2, so $x=\frac{\pi }{6}$ or $\frac{5\pi }{6}$.
• $\sin x=1$: $x=\frac{\pi }{2}$.

The solutions are $x=\frac{\pi }{6},\frac{\pi }{2},\frac{5\pi }{6}$.

Other useful moves

• $\sin x=\sqrt{3}\cos x$: divide by $\cos x$ to get $\tan x=\sqrt{3}$ (check $\cos x=0$ is not a solution first).
• ${\sin }^{2}x=\frac{3}{4}$: take both square roots, $\sin x=\pm \frac{\sqrt{3}}{2}$, giving solutions in all four quadrants.
• $\sin x\cos x=\frac{1}{2}\cos x$: move everything to one side and factorise. Dividing by $\cos x$ would lose the solutions of $\cos x=0$.
• If $\sin x=k$ with $|k|>1$, there is no solution.

• Giving answers in degrees when the domain is in radians, or the reverse.
• Forgetting to extend the domain for $2x$ or $3x$, which loses half the solutions.
• Dividing both sides by $\sin x$ or $\cos x$ and losing the solutions where it equals zero.
• Writing ${\sin }^{2}x$ as $\sin x^2$, which means something different.
• Thinking $\sec x=\frac{1}{\sin x}$. It is the other way round: sec goes with cos, and cosec goes with sin.
• In a proof, treating the identity as an equation (moving terms across or doing the same thing to both sides), or ending without stating the result.

• Solve means give every solution in the domain. Sketching the graph of the ratio over the domain is a quick check of how many there should be.
• Prove or show that questions are marked on the logical chain: one identity per line, with LHS at the start and RHS at the end.
• Give exact answers (in terms of $\pi$) unless the question asks for decimals. If the related angle is not a standard one, use the calculator and round only at the end.
• Read the domain carefully: $0\le x\le 2\pi$ includes both endpoints, and solutions like $x=0$ or $x=2\pi$ count.

• Trigonometry and measure of angles (Year 11): exact values, related angles and the unit circle definitions are the starting point here.
• Further graph transformations and modelling (Year 12): solving $f(x)=k$ for transformed trig functions uses the same domain-shifting method.
• Differential and integral calculus (Year 12): finding stationary points of trig functions means solving trig equations, and identities help simplify derivatives and integrals.