A surveyor is checking the height of a vertical communication tower TF that stands on level horizontal ground. She stands at a point P on the ground, 28 m from the foot F of the tower. From P, the angle of elevation of the top T of the tower is 37°. Because the tower TF is verti…

(a) TF ≈ 21.1 m (b) PT ≈ 35 m

Trigonometry (SOHCAHTOA): O-Level / IGCSE Maths (0580)

Syllabus C6.2, E6.2 · Strand 6 Trigonometry

Questions: 3
Total marks: 13
Tier mix: 1 Core · 2 Extended

This topic is about working with right-angled triangles: given some sides and angles, you use trigonometry to find the ones you do not yet know. The three ratios are sine, cosine and tangent, and they connect an acute angle to two of the triangle’s sides. Relative to a chosen angle $\theta$ , the sides are named the opposite, the adjacent, and the hypotenuse (the longest side, always facing the right angle). The mnemonic SOHCAHTOA records the three relationships compactly: $\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$ , $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$ , and $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$ .

The method has two main directions. To find an unknown side, label the sides relative to the known angle, pick the ratio that uses the side you want and the side you already have, then rearrange and evaluate, for example $x = 12\sin 35^\circ$ . To find an unknown angle, form the correct ratio from two known sides and apply the inverse function, such as $\theta = \tan^{-1}\!\left(\dfrac{\text{opp}}{\text{adj}}\right)$ . Make sure your calculator is in degree mode, and keep enough accuracy in intermediate steps before rounding the final answer.

A common application is angles of elevation and depression: the angle measured up from the horizontal to an object above you (elevation), or down from the horizontal to an object below you (depression). These problems are solved by sketching the right-angled triangle hidden in the situation and choosing the appropriate ratio. The worked examples below are original, written from the syllabus objective to show these exam-style techniques, with each worked solution laid out step by step.

Question 1

Structured Core 4 marks

A surveyor is checking the height of a vertical communication tower $TF$ that stands on level horizontal ground.

She stands at a point $P$ on the ground, $28\text{ m}$ from the foot $F$ of the tower. From $P$ , the angle of elevation of the top $T$ of the tower is $37^\circ$ .

Because the tower $TF$ is vertical and $P$ and $F$ are on the same horizontal level, triangle $PFT$ is right-angled at $F$ .

(a) Calculate the height of the tower, $TF$ . Give your answer correct to 1 decimal place. [2]

(b) Calculate the distance $PT$ from the surveyor to the top of the tower. Give your answer correct to the nearest metre. [2]

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Question 2

Structured Extended 6 marks

A vertical lighthouse stands on a horizontal sea wall. The lamp room window $W$ is $48\text{ m}$ vertically above sea level. A lighthouse keeper at $W$ watches two fishing boats, $A$ and $B$ , which are on the sea on the same side of the lighthouse, in line with its base $F$ .

Boat $A$ is at a horizontal distance of $95\text{ m}$ from the foot of the lighthouse $F$ . Boat $B$ is closer in, at a horizontal distance of $60\text{ m}$ from $F$ .

(All distances are measured in a vertical plane. You may assume the line $FW$ is vertical and the sea surface through $F$ , $B$ , $A$ is horizontal.)

(a) Calculate the angle of depression of boat $A$ from the window $W$ . Give your answer correct to $1$ decimal place. [3]

(b) Calculate the angle of depression of boat $B$ from the window $W$ . Give your answer correct to $1$ decimal place. [2]

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Question 3

Multiple choice Extended 3 marks

A surveyor stands on level ground at point $P$ , which is $24\text{ m}$ from the base $B$ of a vertical flagpole. From $P$ the angle of elevation to the top $T$ of the flagpole is $37^\circ$ . Triangle $PBT$ is right-angled at $B$ . Calculate the height $BT$ of the flagpole, giving your answer correct to $3$ significant figures.

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