Math Core

The Pythagorean theorem connects the three sides of a right triangle, but it says nothing about the angles. Trigonometry fills that gap. Its first tool, the tangent ratio, lets you find a height you can't reach, like a tree or a cliff, from a distance you can measure and an angle.

Naming the sides from an angle

Pick one of the acute angles of a right triangle and call it $\theta$ (the Greek letter theta). Relative to $\theta$ , the three sides get names:

The hypotenuse is across from the right angle, as always.
The opposite side is across from $\theta$ . It doesn't touch $\theta$ at all.
The adjacent side is the leg that forms $\theta$ together with the hypotenuse.

Side names are relative to the angle θ. The opposite side does not touch θ.

The names depend on which angle you're looking from. The leg that is opposite one acute angle is adjacent to the other one.

Why a ratio of sides depends only on the angle

Draw two right triangles that both have a $35^\circ$ angle. They also both have a $90^\circ$ angle, so they are similar by AA. One might be tiny and the other huge, but their sides are proportional. That means the ratio

$\frac{\text{opposite}}{\text{adjacent}}$

is the same in both triangles. It doesn't depend on the size of the triangle, only on the angle. For $35^\circ$ it's about $0.7002$ , for any right triangle you draw. That fixed number gets a name.

Definition

Tangent

In a right triangle with acute angle $\theta$ ,

$\tan \theta = \frac{\text{length of the side opposite } \theta}{\text{length of the side adjacent to } \theta}.$

Worked example: Writing tangent ratios

In right triangle $ABC$ , the right angle is at $C$ , $BC = 5$ and $AC = 12$ . Find $\tan A$ and $\tan B$ .

From $A$ : the opposite side is $BC = 5$ and the adjacent side is $AC = 12$ , so

$\tan A = \frac{5}{12}.$

From $B$ : the opposite side is $AC = 12$ and the adjacent side is $BC = 5$ , so

$\tan B = \frac{12}{5}.$

The two tangents are reciprocals. That always happens for the two acute angles of a right triangle, because their opposite and adjacent sides trade places.

Finding a side

Your calculator knows the tangent of every angle. Make sure it's in degree mode (look for DEG on the screen); otherwise every answer will be wrong. As a quick check, $\tan 45^\circ$ should be exactly $1$ .

To find a missing side, write the tangent equation and solve it.

Worked example: The unknown is on top

Find $x$ to the nearest tenth.

The side $x$ is opposite the $35^\circ$ angle and $10$ is adjacent, so

\begin{aligned} \tan 35^\circ &= \frac{x}{10} \\ x &= 10 \tan 35^\circ \approx 10(0.7002) \approx 7.0 \end{aligned}

Worked example: The unknown is on the bottom

Find $x$ to the nearest tenth.

Now $8$ is opposite the $40^\circ$ angle and $x$ is adjacent:

\begin{aligned} \tan 40^\circ &= \frac{8}{x} \\ x \tan 40^\circ &= 8 \\ x &= \frac{8}{\tan 40^\circ} \approx \frac{8}{0.8391} \approx 9.5 \end{aligned}

Common mistake

When the unknown is in the denominator, a very common mistake is to write $x = 8 \tan 40^\circ$ . Multiply both sides by $x$ first, then divide by $\tan 40^\circ$ . A quick check helps: a $40^\circ$ angle is less than $45^\circ$ , so the opposite side should be shorter than the adjacent side. Here $8 < 9.5$ , which fits.

Angles of elevation

When you look up at something, the angle between your line of sight and the horizontal is the angle of elevation. The ground and a vertical object like a tree or a building form a right angle, so tangent is perfect for these problems: the height is opposite the angle and the ground distance is adjacent.

Worked example: How tall is the tree?

A tree casts a shadow $30$ feet long. The angle of elevation from the tip of the shadow to the top of the tree is $52^\circ$ . How tall is the tree, to the nearest tenth of a foot?

The height $h$ is opposite the $52^\circ$ angle, and the shadow is adjacent:

$h = 30 \tan 52^\circ \approx 30(1.2799) \approx 38.4 \text{ feet}.$

Working backwards: inverse tangent

If you know the two legs, you can find the angle. The inverse tangent, written $\tan^{-1}$ (the $\text{tan}^{-1}$ key on a calculator), answers the question "which angle has this tangent?"

If $\tan \theta = \dfrac{7}{4}$ , then $\theta = \tan^{-1}\left(\dfrac{7}{4}\right) \approx 60.3^\circ$ .

The $-1$ here is not an exponent. $\tan^{-1} x$ means "the angle whose tangent is $x$ ", not $\dfrac{1}{\tan x}$ . You'll use inverse trig functions much more in the lesson on solving right triangles.

Tip

Tangent grows as the angle grows. Below $45^\circ$ the tangent is less than $1$ , at $45^\circ$ it equals $1$ , and above $45^\circ$ it's greater than $1$ . Use this to check whether a side or angle you found is reasonable.

Practice

In right triangle $ABC$ below, the right angle is at $C$ , $AC = 8$ and $BC = 15$ . Use it for the first two problems.

Practice 1

Find $\tan A$ as a fraction.