Cosine

Cosine tells us how much of a right triangle's hypotenuse lies in the direction adjacent to an angle.

Meaning in a right triangle

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

This is CAH in SOH–CAH–TOA. The adjacent side touches $\theta$ but is not the hypotenuse. If both sides touching the angle seem “adjacent,” first identify and exclude the hypotenuse.

Why cosine belongs to the angle

Right triangles sharing the same acute angle are similar. Corresponding side lengths change by the same scale factor, so adjacent divided by hypotenuse stays constant.

In a $5$–$12$–$13$ triangle, viewed from the angle beside the side of length $12$,

Scaling the triangle does not change this fraction, which is why a calculator can give cosine from the angle alone.

When to use cosine

Choose cosine when the sides involved are the adjacent and hypotenuse. If the two useful sides are opposite and hypotenuse, use sine. If they are opposite and adjacent, use tangent.

Always choose the reference angle before labelling adjacent. A side may be adjacent to one acute angle and opposite the other.

Worked example: find the adjacent side

Worked example: find the hypotenuse

A guy wire forms a $63^\circ$ angle with the ground, and its ground anchor is $5.2$ m horizontally from the pole. Let $L$ be the wire length:

Rearrange:

Because the wire is the hypotenuse, it must be longer than $5.2$ m; this is a useful reasonableness check.

Finding an angle with inverse cosine

When adjacent and hypotenuse are known, use inverse cosine after forming the ratio.

For adjacent $=9$ and hypotenuse $=15$,

Inverse cosine returns an angle. The reciprocal $1/\cos\theta$ is a different function called secant.

How cosine changes

For acute angles, cosine lies between $0$ and $1$. As $\theta$ increases toward $90^\circ$, the adjacent side becomes a smaller fraction of the hypotenuse, so cosine decreases.

Important exact values include

Notice the complementary relationship $\cos\theta=\sin(90^\circ-\theta)$ for acute angles.

Connections to coordinates and vectors

On the unit circle, cosine is the $x$-coordinate of a point at angle $\theta$. This extends the definition beyond right triangles and allows negative cosine values on the left half of the circle.

Cosine also measures horizontal components. A force of magnitude $F$ acting at angle $\theta$ above the horizontal has horizontal component $F\cos\theta$. Similar component ideas appear in physics, navigation, and vector mathematics.

Cosine in non-right triangles

The cosine law generalizes the Pythagorean theorem:

It can solve a non-right triangle when three sides are known or when two sides and their included angle are known. Right-triangle cosine and the cosine law are connected, but their setups are different.

Common mistakes

Calling the hypotenuse adjacent. Exclude the side opposite $90^\circ$ before naming adjacent.

Changing the reference angle silently. Relabel the sides whenever the selected angle changes.

Using cosine with opposite and hypotenuse. That pair belongs to sine.

Confusing $\cos^{-1}$ with $1/\cos$. One finds an angle; the other is secant.

Accepting a hypotenuse shorter than a leg. Use geometry to catch an algebra or calculator error.

Quick self-check

• Did I mark the selected acute angle?
• Is the hypotenuse opposite $90^\circ$?
• Is the other relevant side adjacent to $\theta$?
• Does the equation put adjacent over hypotenuse?
• Is the calculator in degree mode, and is the final answer sensible?

Related topics

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