Trigonometry

Trigonometry links the sides and the angles of a right-angled triangle. If you know one side and one of the other angles, you can work out another side. If you know two sides, you can work out an angle.

In the exam you are usually shown a right-angled triangle with an angle and a side marked, and asked to work out a side or an angle, for 2 marks. These questions are on the calculator papers. You round the answer, often correct to 1 decimal place or to 3 significant figures. On the non-calculator paper you may be asked for an exact value, such as \cos 60^\circ, for 1 mark.

Harder questions, for 3 to 5 marks, hide the right-angled triangle inside another shape, such as a trapezium, or need more than one step.

Pythagoras’ theorem uses three sides and no angles. If a question gives you an angle, or asks you for one, use trigonometry instead.

Before you do anything else, label the three sides. Start from the angle you are given, or the angle you want to find, not the right angle.

• The hypotenuse is opposite the right angle. It is the longest side.
• The opposite side is across the triangle from the angle.
• The adjacent side is next to the angle, and is not the hypotenuse.

If you use the other angle instead, the opposite and adjacent sides swap over. The hypotenuse never changes.

Each ratio uses two of the sides. For an angle x:

\sin x=\frac{\text{opposite}}{\text{hypotenuse}}

\cos x=\frac{\text{adjacent}}{\text{hypotenuse}}

\tan x=\frac{\text{opposite}}{\text{adjacent}}

Many people remember these as SOH CAH TOA: sin is opposite over hypotenuse, cos is adjacent over hypotenuse, tan is opposite over adjacent. Since 2022 a formulae sheet has been given in the exam, with these three ratios on it, but learn them anyway.

\sin 35^\circ is just a number. Your calculator gives \sin 35^\circ=0.573576\ldots This means that in every right-angled triangle with a 35^\circ angle, the opposite side is about 0.574 times the hypotenuse, however big the triangle is.

In this diagram the hypotenuse is 1. So the opposite side is \sin 35^\circ, about 0.57, and the adjacent side is \cos 35^\circ, about 0.82. With a bigger angle, the opposite side would be longer and the adjacent side shorter.

Follow the same steps every time: - Label the sides from the angle you know. - Tick the side you know and the side you want. - Choose the ratio that uses both ticked sides. - Write it as an equation, put in the numbers, and work out the side. - Round at the end, as the question asks.

For example, if you know the adjacent side and want the opposite side, the ratio with opposite and adjacent is TOA, so use tan.

If the side you want is on the top of the fraction, multiply. For example, \cos 25^\circ=\frac{x}{8} gives x=8\times\cos 25^\circ

Check your calculator is in degree mode: a small D shows at the top of the screen. To work out 8\times\cos 25^\circ, press 8, ×, cos, 25, close the bracket, then =. You should get 7.250462\ldots

Sometimes the side you want ends up on the bottom of the fraction. This happens when you want the hypotenuse, or when you use tan to find the adjacent side.

For example, \cos 60^\circ=\frac{5}{x}. Multiply both sides by x, then divide both sides by \cos 60^\circ:

x=\frac{5}{\cos 60^\circ}=10

The side you know is divided by the sin, cos or tan of the angle.

So if x is on the top, multiply. If x is on the bottom, divide. Then check: a hypotenuse must come out longer than the other two sides.

To find an angle you need two sides. Label them from the angle you want, choose the ratio in the same way, and write it as a fraction.

Then use the inverse key. On most calculators you press SHIFT and then sin, cos or tan. This gives \sin^{-1}, \cos^{-1} or \tan^{-1}, which turn the ratio back into the angle.

For example, if \tan x=\frac{3}{5}, then x=\tan^{-1}\left(\frac{3}{5}\right)=30.963756\ldots^\circ, which is 31.0^\circ correct to 1 decimal place.

The ratio, 0.6 here, is not the angle. In a right-angled triangle the other two angles are always between 0^\circ and 90^\circ, so check that your answer is too.

On the non-calculator paper you may be asked to write down an exact value, such as \sin 30^\circ. These are not on the formulae sheet, so learn them:

\begin{array}{c|ccc} \text{angle} & \sin & \cos & \tan \\[0.6em] 0^\circ & 0 & 1 & 0 \\[0.6em] 30^\circ & \frac12 & \frac{\sqrt3}{2} & \frac{1}{\sqrt3} \\[0.6em] 45^\circ & \frac{\sqrt2}{2} & \frac{\sqrt2}{2} & 1 \\[0.6em] 60^\circ & \frac{\sqrt3}{2} & \frac12 & \sqrt3 \\[0.6em] 90^\circ & 1 & 0 & \end{array}

Look for the patterns. The cos column is the sin column upside down. And \tan 45^\circ=1, because a 45^\circ right-angled triangle has its opposite and adjacent sides the same length. There is no value of \tan 90^\circ to learn.

You can use them to find a side without a calculator. For example, if \tan 45^\circ=\frac{x}{8}, then x=8\times1=8

• Using a calculator that is not in degree mode. Check that a small D is showing before you start.
• Choosing the wrong ratio. Label the sides from the angle first, then pick the ratio that uses the side you know and the side you want.
• Using Pythagoras or the angles in a triangle instead. When an angle is given or asked for, use sin, cos or tan.
• Putting the side inside the bracket. Close the bracket after the angle: 8\times\cos(25), not \cos(8\times25).
• Writing the fraction upside down. The first side named goes on top: tan is opposite over adjacent, not adjacent over opposite.
• Giving the ratio as the angle. To find an angle, finish with the inverse key, \sin^{-1}, \cos^{-1} or \tan^{-1}.