POP

G5.19 SOHCAHTOA

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01 — Explanation

How do I correctly label a right-angled triangle's sides?

Always label in this order to avoid ambiguity:

Hypotenuse (H): opposite the right angle, always the longest side — fixed regardless of which acute angle you're using.
Opposite (O): directly across from your reference angle.
Adjacent (A): the remaining side, touching the reference angle but not the hypotenuse.

Exam Tip

Practise labelling triangles that are rotated or flipped — if you rely on a mental picture of a "standard" triangle rather than this 3-step rule, O and A will swap whenever the reference angle changes.

02 — Explanation

Which ratio do I use, and how do I remember them?

\sin\theta=\frac{O}{H} \qquad \cos\theta=\frac{A}{H} \qquad \tan\theta=\frac{O}{A}

Write SOHCAHTOA at the top of your working before starting any problem — it's one mnemonic covering all three ratios, and under exam pressure it's easy to misremember which letter goes where.

03 — Explanation

How do I find a missing side?

Label H, O, A relative to the given angle.
Decide which ratio connects the side you have to the side you want.
Substitute and solve.

If the unknown is in the numerator (straightforward): e.g. $\sin36°=\dfrac{x}{10} \Rightarrow x=10\sin36°\approx5.88\text{ cm}$ .

If the unknown is in the denominator (needs an extra step): e.g. $\tan40°=\dfrac{45}{y}$ . Multiply both sides by $y$ first, then divide by $\tan40°$ : $y=\dfrac{45}{\tan40°}\approx53.6\text{ ft}$ .

Exam Tip

Rearranging when the unknown is on the bottom of the fraction is where most algebra slips happen — always do it in the two separate steps above rather than trying to "flip" it in one move.

04 — Explanation

How do I find a missing angle?

Use the inverse functions ( $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ ) — these "undo" the ratio to recover the angle. If $\cos(x)=\dfrac{4.8}{8.9}$ , then $x=\cos^{-1}\left(\dfrac{4.8}{8.9}\right)\approx57.4°$ (1 d.p.).

Exam Tip

$\sin^{-1}$ is not $\dfrac{1}{\sin}$ — the $-1$ denotes an inverse function, not a power. Always check your calculator is in degree mode before starting; an answer calculated in radians will be completely wrong despite correct method.

05 — Explanation

How do angle of elevation and depression problems work?

The angle of elevation is measured upward from the horizontal and sits inside your triangle. The angle of depression is measured downward from the horizontal and sits outside the triangle at the top — but because the horizontal line of sight is parallel to the ground, alternate angles mean the depression angle equals the elevation angle measured from the object below.

Worked Example

From the top of a 45-foot tower, the angle of depression to a fire is $40°$ . Find the distance from the base of the tower to the fire.

By alternate angles, the angle of elevation from the fire to the tower top is also $40°$ .
Relative to this angle: tower height ( $45\text{ ft}$ ) is Opposite; ground distance ( $x$ ) is Adjacent. Use TOA.
$\tan40°=\dfrac{45}{x} \Rightarrow x=\dfrac{45}{\tan40°}\approx53.6\text{ ft}$ .

Exam Tip

Sketching a clear diagram first — labelling the horizontal line of sight explicitly — is the most reliable way to avoid placing the angle of depression inside the triangle by mistake.