Sine

Meaning in a right triangle

For an acute angle $\theta$ in a right triangle,

This is the SOH part of SOH–CAH–TOA. The side called opposite depends on the chosen angle. The hypotenuse does not: it is always across from the $90^\circ$ angle.

Worked example: find the opposite side

Worked example: find the hypotenuse

A kite is $24$ m vertically above the holder, and the taut string makes a $55^\circ$ angle with the ground. Ignoring the holder's height, let $L$ be the string length:

Multiply by $L$, then divide by $\sin55^\circ$:

Dividing is necessary because the unknown appears in the denominator.

Finding an angle with inverse sine

If the opposite and hypotenuse are known, first form their ratio and then use inverse sine.

Suppose opposite $=8$ and hypotenuse $=17$:

On a calculator, $\sin^{-1}$ usually appears above the sine key. It returns an angle; it is not the same as $1/\sin\theta$.

Calculator and notation care

Use degree mode for problems whose angles include a degree symbol. Enter the full ratio inside inverse sine, preferably with parentheses. Keep the unrounded value until the end, then report the requested precision and units.

The notation $\sin^2\theta$ means $(\sin\theta)^2$. It does not mean $\sin(\theta^2)$.

Common mistakes

Using the adjacent side. Sine uses opposite over hypotenuse.

Misidentifying the hypotenuse. Find the side opposite $90^\circ$, not merely the side drawn diagonally.

Multiplying when the unknown is the denominator. Rearrange the equation carefully before calculating.

Confusing inverse and reciprocal. $\sin^{-1}$ finds an angle; $1/\sin$ is cosecant.

Rounding the ratio too early. Keep calculator precision through the final step.