Cosine

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.

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.

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?