SOHCAHTOA — Finding Missing Sides

SOHCAHTOA is a mnemonic for the three trigonometric ratios in a right-angled triangle. It is used to find missing side lengths when an angle and one side are known.

What is SOHCAHTOA?

SOHCAHTOA is a memory aid for three ratios that link the angles and sides of a right-angled triangle.

Mnemonic	Ratio	Formula
SOH	Sine	$\sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}$
CAH	Cosine	$\cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}$
TOA	Tangent	$\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}$

Labelling the Triangle

Always label the sides relative to the angle you are working with (not the right angle):

Hypotenuse — the longest side, always opposite the 90° angle
Opposite — the side directly across from the angle $\theta$
Adjacent — the side next to the angle $\theta$ (not the hypotenuse)

Using SOHCAHTOA to Find a Missing Side

Label all sides (Hyp, Opp, Adj) relative to the known angle.
Choose the ratio that uses the known side AND the side you want to find.
Substitute into the formula and solve.

Find the opposite side (sin rule)

Angle = 30°, Hypotenuse = 10 cm. Find the Opposite.

We know Hyp and want Opp → SOH: $\sin 30° = \frac{\text{Opp}}{10}$

$\text{Opp} = 10 \times \sin 30° = 10 \times 0.5 = 5 \text{ cm}$

Find the adjacent side (cos rule)

Angle = 40°, Hypotenuse = 8 cm. Find the Adjacent.

CAH: $\cos 40° = \frac{\text{Adj}}{8}$

$\text{Adj} = 8 \times \cos 40° \approx 8 \times 0.766 = 6.13 \text{ cm}$

Find the hypotenuse (sin rule, hyp unknown)

Angle = 25°, Opposite = 6 cm. Find the Hypotenuse.

$\sin 25° = \frac{6}{\text{Hyp}}$

$\text{Hyp} = \frac{6}{\sin 25°} \approx \frac{6}{0.423} \approx 14.2 \text{ cm}$

Tip

If a side is on top of the fraction you multiply; if it's on the bottom you divide.

Real-World Applications

Real world: surveying — height of a building

A surveyor stands 50 m from the base of a building and measures the angle of elevation to the top as 35°. The 50 m is the adjacent side; the height is opposite. Using TOA:

$\tan 35° = \dfrac{\text{height}}{50}$ → $\text{height} = 50 \times \tan 35° \approx 50 \times 0.700 = 35 \text{ m}$

Real world: wheelchair ramp design

A building regulation states a ramp must not rise more than 1 m over a horizontal distance of 12 m. The maximum ramp angle is:

$\theta = \tan^{-1}\!\left(\dfrac{1}{12}\right) \approx 4.8°$

A designer uses SOHCAHTOA to check that $\sin 4.8° \times 12 \approx 1$ m rise — confirming compliance.

Real world: finding the length of a roof rafter

A roof rises 2.5 m over a horizontal span of 5 m. The rafter (hypotenuse) length is found using the angle:

$\tan\theta = \dfrac{2.5}{5} = 0.5$ → $\theta \approx 26.6°$

$\text{Hyp} = \dfrac{2.5}{\sin 26.6°} \approx \dfrac{2.5}{0.447} \approx 5.6 \text{ m}$

Tip

Real-world shortcut: Always identify which two sides are involved, pick the correct ratio (SOH, CAH, or TOA), then rearrange: multiply if the unknown is on top, divide if it is on the bottom.

Practice Problems

Find the opposite side. Angle = 50°, Hyp = 12 cm.

Show answer

sin 50° = Opp/12 → Opp = 12 × sin 50° ≈ 12 × 0.766 ≈ 9.2 cm

Find the adjacent side. Angle = 60°, Hyp = 20 cm.

Show answer

cos 60° = Adj/20 → Adj = 20 × 0.5 = 10 cm

Find the hypotenuse. Angle = 35°, Opp = 9 cm.

Show answer

sin 35° = 9/Hyp → Hyp = 9/sin 35° ≈ 9/0.574 ≈ 15.7 cm