A surveyor stands on level ground at point P, which is 24 m from the base B of a vertical flagpole. From P the angle of elevation to the top T of the flagpole is 37°. Triangle PBT is right-angled at B. Calculate the height BT of the flagpole, giving your answer correct to 3 sign…

IGCSE Trigonometry MCQ: Choosing the Correct Ratio (SOHCAHTOA)

Step 1: Identify the known and unknown sides relative to the angle

Place the $37^\circ$ angle at point $P$ . In the right-angled triangle $PBT$ (right angle at $B$ ):

The horizontal distance $PB = 24\text{ m}$ is adjacent to the $37^\circ$ angle.
The height $BT$ (what we want) is opposite the $37^\circ$ angle.
The hypotenuse is $PT$ , but we are not told it and are not asked for it.

Step 2: Choose the correct trigonometric ratio

We have the opposite (unknown) and the adjacent (known), so the ratio linking them is tangent (the “T-O-A” part of SOHCAHTOA):

$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

$\tan 37^\circ = \frac{BT}{24}$

Sine or cosine would require the hypotenuse, which we do not have, so they are the wrong choice here.

Step 3: Solve for the unknown side

Multiply both sides by $24$ :

$BT = 24 \times \tan 37^\circ$

$BT = 24 \times 0.75355\ldots = 18.085\ldots$

$BT \approx \boxed{18.1\text{ m}\ (3\text{ s.f.})}$

Why the other values are wrong

Choice	What was done	Error
$14.4\text{ m}$	$24\sin 37^\circ$	Treated $24$ as the hypotenuse (used SOH wrongly)
$19.2\text{ m}$	$24\cos 37^\circ$	Swapped opposite and adjacent (used CAH wrongly)
$31.9\text{ m}$	$24 \div \tan 37^\circ$	Inverted the ratio ( $\tan 37^\circ = \tfrac{24}{BT}$ )

Quick sanity check

Since $37^\circ < 45^\circ$ , the opposite side $BT$ should be shorter than the adjacent side $24\text{ m}$ . Our answer $18.1\text{ m} < 24\text{ m}$ , which is consistent. (Both $19.2\text{ m}$ and $31.9\text{ m}$ fail this check, and $14.4\text{ m}$ requires the wrong ratio.)

Final answer: $BT \approx 18.1\text{ m}$ , option B.