Right Triangle Trigonometry

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.

SOH–CAH–TOA

The three primary ratios are

The memory aid SOH–CAH–TOA matches each function with its two sides. It is not a substitute for a labelled sketch: label $\theta$, H, O, and A before choosing a ratio.

Choosing the right ratio

Ignore any side for which you have neither a value nor a target. Then choose the ratio containing the two sides that remain.

If two sides are known and the angle is unknown, use the same ratio and then an inverse trig function.

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,

Use inverse tangent:

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

Calculator setup and rounding

Ontario high-school geometry questions normally use degrees, so confirm that the calculator displays DEG, not RAD. A correct method in the wrong angle mode can produce a completely different answer.

Carry several decimal places during the calculation. Round only once, according to the question's requested precision. Include units for a length and a degree symbol for an angle.

Check whether the answer is sensible

A right-triangle answer should agree with the picture and basic geometry:

• the hypotenuse must be the longest side;
• both acute angles must lie between $0^\circ$ and $90^\circ$;
• the two acute angles must add to $90^\circ$;
• a side opposite a larger angle should be longer than a side opposite a smaller angle.

The Pythagorean theorem can check a side-length result when enough lengths are known.

Applications and modelling

Right triangle trigonometry is used for heights, shadows, ladders, roof pitch, navigation, ramps, sight lines, construction, and indirect distances. A modelling question may include extra information, so draw only the triangle that connects the requested quantity to known measurements.

An angle of elevation is measured upward from a horizontal line of sight. An angle of depression is measured downward. Because horizontal lines are parallel, these often create equal alternate interior angles in a diagram.

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?

Related topics

• Sine
• Cosine
• Tangent
• Pythagorean Theorem
• Similar Triangles