Sine

Sine connects an acute angle to the fraction of the hypotenuse that appears opposite that angle.

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.

Why the ratio stays constant

Any two right triangles with the same acute angle are similar. Their corresponding side lengths may scale up or down, but the quotient opposite divided by hypotenuse stays equal. Sine is therefore a property of the angle, not of one particular triangle.

For example, if a $5$–$12$–$13$ triangle is viewed from the angle opposite the side of length $5$, then

A triangle twice as large would give $10/26$, the same ratio.

When to use sine

Use sine in a right triangle when the relevant sides are the opposite and hypotenuse. You might know one side and an angle and need the other side, or know both sides and need the angle.

If the known and target sides are adjacent and hypotenuse, use cosine. If they are opposite and adjacent, use tangent.

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 $1/\sin\theta$.

How sine changes

For acute angles, sine is between $0$ and $1$. As $\theta$ grows from near $0^\circ$ to near $90^\circ$, the opposite side becomes a larger fraction of the hypotenuse, so $\sin\theta$ increases.

Useful benchmark values are

These values help estimate whether a decimal answer is plausible.

Beyond right triangles

On the unit circle, sine is the $y$-coordinate of a point reached by rotating through an angle. This definition extends sine beyond acute angles and explains why sine can be negative and why its graph repeats.

Later courses use sine to model tides, sound, seasonal daylight, circular motion, and other repeating phenomena. The right-triangle ratio is the first part of that larger idea.

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.

Quick self-check

• Is the triangle right-angled?
• Which side is opposite the selected angle?
• Which side is the hypotenuse?
• Am I finding a side with sine or an angle with inverse sine?
• Is the result consistent with hypotenuse being longest?

Related topics

• Right Triangle Trigonometry
• Cosine
• Tangent
• Unit Circle
• Sine Law