Math Core

To solve a right triangle means to find all three sides and all three angles. With the Pythagorean theorem, the three trig ratios and their inverses, you can do it from surprisingly little information: any two sides, or one side and one acute angle.

Inverse trig functions

The trig ratios turn an angle into a number. The inverse trig functions turn the number back into an angle.

Definition

Inverse sine, cosine and tangent

For an acute angle $\theta$ :

If $\sin \theta = x$ , then $\theta = \sin^{-1} x$ .
If $\cos \theta = x$ , then $\theta = \cos^{-1} x$ .
If $\tan \theta = x$ , then $\theta = \tan^{-1} x$ .

Read $\sin^{-1} x$ as "inverse sine of $x$ " or "the angle whose sine is $x$ ." Some books write $\arcsin x$ instead.

For example, $\sin 30^\circ = 0.5$ , so $\sin^{-1}(0.5) = 30^\circ$ . On a calculator, these are usually the second functions of the sin, cos and tan keys. As always, stay in degree mode.

A strategy for any right triangle

You already know one angle ( $90^\circ$ ). Beyond that, every problem falls into one of two cases.

Solving a right triangle

Given two sides:

Find the third side with the Pythagorean theorem.
Find one acute angle with an inverse trig function.
Find the other acute angle by subtracting from $90^\circ$ .

Given one side and one acute angle:

Find the other acute angle by subtracting from $90^\circ$ .
Find each missing side with a trig ratio that uses the side you were given.

Whenever you can, compute from the numbers you were given, not from values you rounded along the way. Rounding early lets small errors build up.

Worked example: Given two legs

Solve right triangle $ABC$ with legs $AC = 14$ and $BC = 9$ . Round to the nearest tenth.

Hypotenuse: $c = \sqrt{14^2 + 9^2} = \sqrt{196 + 81} = \sqrt{277} \approx 16.6$ .

Angle A: from $A$ , $9$ is opposite and $14$ is adjacent, so

$A = \tan^{-1}\left(\frac{9}{14}\right) \approx 32.7^\circ.$

Angle B: $B = 90^\circ - 32.7^\circ = 57.3^\circ$ .

So $c \approx 16.6$ , $A \approx 32.7^\circ$ and $B \approx 57.3^\circ$ . Check: the biggest acute angle, $B$ , is opposite the longer leg, $14$ . ✓

Worked example: Given the hypotenuse and an angle

A right triangle has hypotenuse $20$ and an acute angle $A = 36^\circ$ . Solve the triangle. Round sides to the nearest tenth.

Other angle: $B = 90^\circ - 36^\circ = 54^\circ$ .

Leg opposite A: $a = 20 \sin 36^\circ \approx 11.8$ .

Leg adjacent to A: $b = 20 \cos 36^\circ \approx 16.2$ .

Both legs came straight from the given hypotenuse, so neither depends on a rounded value. Check with the Pythagorean theorem: $11.76^2 + 16.18^2 \approx 138.3 + 261.8 = 400.1 \approx 20^2$ . ✓

Worked example: Finding an angle with inverse sine

A wheelchair ramp is $12$ feet long and rises $1.5$ feet. What angle does it make with the ground, to the nearest tenth of a degree?

The ramp is the hypotenuse and the rise is opposite the angle, so use sine:

$\theta = \sin^{-1}\left(\frac{1.5}{12}\right) = \sin^{-1}(0.125) \approx 7.2^\circ.$

Angles of elevation and depression

When you look up from the horizontal, the angle is an angle of elevation. When you look down from the horizontal, it's an angle of depression. Both are always measured from a horizontal line, never from a vertical one.

The angle of depression from the top (measured down from the dashed horizontal) equals the angle of elevation from the boat.

In the picture, the dashed horizontal line at the top of the lighthouse is parallel to the ground. The line of sight is a transversal, so the angle of depression and the angle of elevation are alternate interior angles, and they're congruent. That means you can put the angle of depression inside the triangle at the bottom, where it's easy to use.

Worked example: Spotting a boat

From the top of a $45$ -meter lighthouse, the angle of depression to a boat is $12^\circ$ . How far is the boat from the base of the lighthouse, to the nearest tenth of a meter?

The angle of elevation from the boat is also $12^\circ$ . From the boat, the height $45$ is opposite and the distance $d$ is adjacent:

\begin{aligned} \tan 12^\circ &= \frac{45}{d} \\ d &= \frac{45}{\tan 12^\circ} \approx 211.7 \text{ meters} \end{aligned}

Common mistake

Don't put the angle of depression between the line of sight and the vertical side (the lighthouse). It's measured from the horizontal. If you use it at the top of the triangle as if it were the angle inside, you'll use $12^\circ$ where the actual interior angle is $78^\circ$ and get a very wrong answer.

Tip

After solving, do a quick size check: the largest side should be across from the largest angle, and the two acute angles should add up to exactly $90^\circ$ .

Practice

Practice 1

In a right triangle, the leg opposite angle $A$ is $5$ and the hypotenuse is $13$ . Find $A$ to the nearest tenth of a degree.