In the diagram, QPR = 30^{} and |QR| = 5\,{cm}. [Take 30^{} = 1/2 ]. Find the length of {PQ}.

SOHCAHTOA is a mnemonic that gives us an easy way to remember the three main trigonometric ratios They are sine (\sin), cosine (cos) and tangent (tan).

SOH

\sin(\theta) = \frac{\text{Opposite}}{\text{Hypothenuse}}

CAH

\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypothenuse}}

TOA

\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

The longest side (side opposite/facing the right-angle) is the hypotenuse

\sin(\theta) = \frac{\text{Opposite}}{\text{Hypothenuse}}

Opposite side of

30^{\circ}

is 5 cm and hypothenuse side is

\overline{PQ}

\sin 30^{\circ} = \frac{5\,\text{cm}}{|PQ|}

But

\sin 30^{\circ} = \frac{1}{2}

\frac{1}{2} = \frac{5\,\text{cm}}{|PQ|}

Cross multiply

1 \times \overline{PQ} = 2 \times 5\,\text{cm}

\overline{PQ} = 10\,\text{cm}

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