Trigonometry in the Right Triangle

How do you find a side length when the angle is known?

Quick answer

To calculate a side length when an angle is known, choose the trigonometric ratio connecting your known side and desired side: sine for opposite and hypotenuse, cosine for adjacent and hypotenuse, or tangent for opposite and adjacent. Set up the equation, substitute the values, and solve for the unknown side.

The topic explained

Pick the ratio connecting your two sides

Finding a triangle side length when an angle is given comes down to naming the two sides you care about relative to that acute angle: the side you already have, and the side you want to find. Once you name them, the formula picks itself.

Label your triangle with three words: hypotenuse (the side facing the right angle), opposite (the side directly across from your known angle), and adjacent (the side next to your angle that is not the hypotenuse). If your two sides are opposite and hypotenuse, use sine. If they are adjacent and hypotenuse, use cosine. If they are both legs, use tangent.

Solve for the unknown side

Write your chosen equation and substitute the numbers. The angle stays inside the trigonometric function until you calculate its decimal value.

If your unknown side sits in the numerator, multiply both sides by the denominator. For example, if sin(30°) = x / 10, then x = 10 · sin(30°) = 10 · 0.5 = 5. If your unknown sits in the denominator, multiply to bring it out, then divide by the trigonometric value. For instance, if cos(45°) = 8 / x, then x = 8 / cos(45°).

Two quick checks to catch mistakes

First, verify that your calculator is in degree mode. In degree mode, sin(30°) equals 0.5. If your calculator outputs -0.988, it is set to radians.

Second, check side sizes. The hypotenuse must always be the longest side in a right triangle. If you calculate a hypotenuse that is shorter than either leg, you either inverted your ratio or divided when you needed to multiply.

Free printable worksheet

Finding Right Triangle Side Lengths

For 10th grade geometry students. Check that your scientific calculator is set to degree mode before solving, and round answers to two decimal places.

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Finding Right Triangle Side LengthsName:

In a right triangle, the hypotenuse is 10 and one acute angle is 30°. Find the side opposite this angle.
In a right triangle, the hypotenuse is 10 and one acute angle is 30°. Find the side adjacent to this angle.
The leg adjacent to a 40° angle measures 6. Find the length of the opposite leg.
The side opposite a 25° angle measures 7. Find the hypotenuse.
A right triangle has an angle of 55° and an adjacent leg of length 12. Find the hypotenuse.
A 15-foot ladder leans against a vertical wall, making a 70° angle with the flat ground. How high up the wall does the ladder reach?

Common questions

How do I know whether to multiply or divide when isolating the side?

Look at where your variable is located in the fraction. If the unknown side is in the numerator on top, multiply the denominator by the trig ratio. If the unknown is in the denominator on the bottom, swap it with the trig ratio: divide the numerator by the trig ratio.

Why does my calculator give a negative number for sine or cosine?

Your calculator is almost certainly in radian mode instead of degree mode. Acute angles in right triangles are between 0° and 90°, where sine, cosine, and tangent are always positive. Switch your calculator mode to DEG.

Can I use the Pythagorean theorem instead to find a missing side?

You can only use the Pythagorean theorem if you already know two side lengths. When you only know one side length and one acute angle, you must use a trigonometric ratio first.

What is the next topic after finding missing sides?

The next step is finding missing angles when two sides are given, using inverse trigonometric functions (arcsin, arccos, and arctan).

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