Math Core

Tangent uses only the two legs. But often the hypotenuse is the side you know or want: the length of a ladder, a ramp or a rope. Sine and cosine are the ratios that bring the hypotenuse in, and together with tangent they complete the basic toolkit of right-triangle trigonometry.

Three ratios

For the same reason as tangent (all right triangles with a given acute angle are similar), any ratio of two sides depends only on the angle. With three sides there are three main ratios.

Definition

Sine, cosine and tangent

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

\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan \theta = \frac{\text{opposite}}{\text{adjacent}}

Many students remember these with the word SOH-CAH-TOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.

Because the hypotenuse is the longest side, the sine and cosine of an acute angle are always between $0$ and $1$ . If you ever get a sine of $1.3$ , you've put the hypotenuse on top by mistake.

Worked example: All three ratios

In right triangle $ABC$ , the right angle is at $C$ , $BC = 7$ , $AC = 24$ and $AB = 25$ . Find $\sin A$ , $\cos A$ and $\tan A$ .

From $A$ : opposite $= BC = 7$ , adjacent $= AC = 24$ , hypotenuse $= AB = 25$ .

$\sin A = \frac{7}{25} \qquad \cos A = \frac{24}{25} \qquad \tan A = \frac{7}{24}$

Complementary angles

Here is the triangle with standard labels: side $a$ is opposite angle $A$ , side $b$ is opposite angle $B$ , and $c$ is the hypotenuse.

Side a is opposite angle A, side b is opposite angle B, and c is the hypotenuse.

Write the ratios from both acute angles:

$\sin A = \frac{a}{c} = \cos B \qquad\qquad \cos A = \frac{b}{c} = \sin B$

The side opposite $A$ is adjacent to $B$ , so the sine of one angle is the cosine of the other. The two acute angles of a right triangle add up to $90^\circ$ (they are complementary), so $B = 90^\circ - A$ . That's where the name cosine comes from: it's the sine of the complement.

Cofunction relationship

For any acute angle $\theta$ ,

$\sin \theta = \cos(90^\circ - \theta) \qquad\text{and}\qquad \cos \theta = \sin(90^\circ - \theta).$

For example, $\sin 20^\circ = \cos 70^\circ$ (both are about $0.342$ ). Check it on your calculator.

Two more connections

Since $\sin A = \dfrac{a}{c}$ and $\cos A = \dfrac{b}{c}$ , dividing gives $\dfrac{\sin A}{\cos A} = \dfrac{a}{b} = \tan A$ . And squaring and adding gives

$(\sin A)^2 + (\cos A)^2 = \frac{a^2 + b^2}{c^2} = \frac{c^2}{c^2} = 1.$

This is the Pythagorean theorem in disguise. It lets you get cosine from sine (or the reverse) without knowing the triangle.

Exact values from special triangles

The special right triangles from earlier give exact trig values for $30^\circ$ , $45^\circ$ and $60^\circ$ . For example, in a 30°-60°-90° triangle with sides $1$ , $\sqrt{3}$ , $2$ , the side opposite $30^\circ$ is $1$ , so $\sin 30^\circ = \dfrac{1}{2}$ .

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$
$30^\circ$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{3}}{3}$
$45^\circ$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$	$1$
$60^\circ$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$\sqrt{3}$

You don't need to memorize the table if you can sketch the two triangles and read off the ratios.

Finding sides

The steps are the same as with tangent: choose the ratio that uses the side you know and the side you want, write the equation, and solve.

Worked example: Finding the hypotenuse

Find $x$ to the nearest tenth.

The known side, $6$ , is opposite the $32^\circ$ angle, and $x$ is the hypotenuse. Opposite and hypotenuse means sine.

\begin{aligned} \sin 32^\circ &= \frac{6}{x} \\ x &= \frac{6}{\sin 32^\circ} \approx \frac{6}{0.5299} \approx 11.3 \end{aligned}

Worked example: A leaning ladder

A $15$ -foot ladder leans against a wall and makes a $50^\circ$ angle with the ground. How far is the foot of the ladder from the wall, to the nearest tenth of a foot?

The ladder is the hypotenuse. The distance $d$ along the ground is adjacent to the $50^\circ$ angle. Adjacent and hypotenuse means cosine.

$d = 15 \cos 50^\circ \approx 15(0.6428) \approx 9.6 \text{ feet}.$

Worked example: Using the cofunction relationship

Find the acute angle $x$ if $\sin x = \cos 25^\circ$ .

Sine and cosine of complementary angles are equal, so $x = 90^\circ - 25^\circ = 65^\circ$ .

Common mistake

The words "opposite" and "adjacent" only make sense from a particular angle. Before choosing a ratio, put your finger on the angle you're using. The side it doesn't touch is opposite; the leg it does touch is adjacent. Mixing up angles is the most common source of wrong answers here.

Tip

Check a side you found against the others: the hypotenuse has to be longer than each leg. In the first example above, $x \approx 11.3$ is longer than $6$ , so it's plausible.

Practice

In right triangle $ABC$ , the right angle is at $C$ , $AC = 20$ , $BC = 21$ and $AB = 29$ . Use it for the first two problems.

Practice 1

Find $\sin A$ as a fraction.