GCSE Maths revision · Geometry and measures

Trigonometry (SOH CAH TOA)

Foundation and Higher

The idea

In a right-angled triangle, $\sin\theta = \frac{O}{H}$ , $\cos\theta = \frac{A}{H}$ and $\tan\theta = \frac{O}{A}$ . Opposite and adjacent are named from the angle you are using.

Label the sides O, A and H from the angle.
Choose the ratio that uses the side you know and the side you need.
For a side, rearrange; for an angle, use the inverse: $\theta = \tan^{-1}\!\left(\frac{O}{A}\right)$ .

Watch out Re-label O and A if you move to the other angle.

See it worked

Worked exampleNon-calculator2 marks

Triangle $ABC$ has a right angle at $B$ . Angle $BAC$ is $\theta$ .

(a) Which side is opposite the angle $\theta$ ?

(b) You know $AB$ and $BC$ and want $\theta$ . Which ratio should you use?

(a) The opposite side is the one that does not touch the angle's vertex $A$ .
(b) Opposite and adjacent are known: tan $=$ opposite $\div$ adjacent.

Answer (a) $BC$ ; (b) $\tan \theta = \frac{BC}{AB}$

Your turn

Question 1Calculator allowed2 marks

In triangle $PQR$ , angle $Q = 90^\circ$ , $PR = 12$ cm and angle $P = 35^\circ$ .

Work out the length of $QR$ . Give your answer correct to 3 significant figures.

Question 2Calculator allowed4 marks

The diagram shows a vertical tower on level ground. From a point 9 m from the foot of the tower, the angle of elevation of the top is $35^\circ$ .

(a) Calculate the height of the tower, to 2 decimal places.

(b) The observer walks to twice the original distance. Calculate the new angle of elevation to 1 decimal place.

Question 3Calculator allowed3 marks

The diagram shows a vertical mast BT on horizontal ground.

P and Q are points on the ground in a straight line with B.

Work out the height of the mast.

Give your answer correct to 1 decimal place.

You must show your working.

In past papers

90 official questions in 87 papers since 2017. The latest:

All GCSE Maths past papers

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