Right Triangle Trigonometry

Right triangle trigonometry connects an acute angle to the side-length ratios sine, cosine, and tangent.

Relative to θ, sine, cosine, and tangent compare opposite, adjacent, and hypotenuse sides. A right triangle has one acute angle labelled theta. The vertical side is opposite theta, the horizontal leg is adjacent, and the sloping longest side is the hypotenuse; a square marks the ninety-degree angle.

Trigonometry lets us find a missing side or angle in a right triangle from surprisingly little information.

The big idea

Every right triangle has one $90^\circ$ angle. Once another acute angle is fixed, the triangle's shape is fixed too: larger and smaller copies have the same side-length ratios. Those constant ratios are called sine, cosine, and tangent.

This means a surveyor can use an angle and one measured distance to estimate a height that would be difficult to measure directly. The mathematics works because similar right triangles keep the same proportions.

Name the sides relative to the angle

Choose the acute reference angle $\theta$ before naming the sides.

The hypotenuse is opposite the $90^\circ$ angle and is always the longest side.
The opposite side is directly across from $\theta$.
The adjacent side touches $\theta$ but is not the hypotenuse.

The hypotenuse keeps its name, but opposite and adjacent switch when the reference angle changes. Marking the angle first prevents most ratio errors.

Worked example: find a side

Keeping the unrounded calculator value until the final line protects accuracy.

Worked example: find an angle

Suppose a ramp rises $0.72$ m over a horizontal run of $9.0$ m. If $\theta$ is its angle with the ground,

\tan\theta=\frac{0.72}{9.0}=0.08.

Use inverse tangent:

\theta=\tan^{-1}(0.08)\approx4.57^\circ.

The superscript $-1$ means “inverse function” here, not reciprocal. On many calculators, use 2nd or shift followed by tan.

Common mistakes

Naming sides before choosing the angle. Opposite and adjacent are relative to $\theta$.

Calling the adjacent side the hypotenuse. The hypotenuse is always across from $90^\circ$.

Using a ratio because it looks familiar. Select the ratio containing both the known side and the target.

Leaving the calculator in radians. Look for DEG before evaluating.

Using inverse trig to find a side. Inverse functions find an angle from a ratio; ordinary trig functions help find sides.

Quick self-check

Did I mark the right angle and reference angle?
Did I label hypotenuse, opposite, and adjacent relative to that angle?
Does my ratio contain the known value and the unknown?
Is my calculator in degree mode?
Did I keep full precision and add units?
Is the result geometrically possible?