Sin, Cos, Tan — Explained

Sine, cosine, and tangent compare side lengths relative to one chosen angle in a right triangle. If you understand which side is opposite, adjacent, and hypotenuse, the three ratios become much easier to use.

If $\theta$ is an acute angle in a right triangle, then

$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$

This is the idea behind SOHCAHTOA. The shortcut helps, but the main point is simpler: each trig function is a ratio attached to one angle, not a property of a side by itself.

What Sin, Cos, And Tan Mean In A Right Triangle

Pick one acute angle $\theta$ in a right triangle.

• The opposite side is across from $\theta$.
• The adjacent side is next to $\theta$, but it is not the hypotenuse.
• The hypotenuse is the longest side, opposite the right angle.

Once those labels are fixed, the trig ratios tell you different comparisons.

• $\sin \theta$ compares opposite to hypotenuse.
• $\cos \theta$ compares adjacent to hypotenuse.
• $\tan \theta$ compares opposite to adjacent.

If you switch to the other acute angle, opposite and adjacent switch too. That is why the same triangle gives different sine, cosine, and tangent values for its two acute angles.

One more useful fact: for a fixed angle, these ratios stay the same even if the triangle is scaled up or down. Similar triangles keep the same angle ratios.

Worked Example With A 3-4-5 Triangle

Suppose a right triangle has side lengths $3$, $4$, and $5$. Let $\theta$ be the acute angle opposite the side of length $3$.

Then:

• opposite $= 3$
• adjacent $= 4$
• hypotenuse $= 5$

So

$$\sin \theta = \frac{3}{5}, \quad \cos \theta = \frac{4}{5}, \quad \tan \theta = \frac{3}{4}$$

This example shows the pattern clearly. Sine and cosine both use the hypotenuse. Tangent does not; it compares the two legs, so it is often useful when you want a sense of steepness.

When To Use Sine, Cosine, Or Tangent

Use these ratios when a problem connects an angle to side lengths in a right triangle.

• Use $\sin \theta$ when the sides you care about are opposite and hypotenuse.
• Use $\cos \theta$ when the sides you care about are adjacent and hypotenuse.
• Use $\tan \theta$ when the sides you care about are opposite and adjacent.

If you know one side and one acute angle, trig often lets you find another side. If you know side ratios, inverse trig functions can help recover the angle.

How The Unit Circle Extends The Same Idea

The right-triangle definitions above apply directly to acute angles in a right triangle. For angles larger than $90^\circ$, negative angles, or full rotations, trigonometry extends the same functions using the unit circle.

On the unit circle, the point for angle $\theta$ is

$$(\cos \theta, \sin \theta)$$

and tangent is still the ratio

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

when $\cos \theta \ne 0$.

So on the unit circle, cosine is the $x$-coordinate and sine is the $y$-coordinate. This is why the same trig names keep working even when there is no right triangle drawn.

Common Mistakes With Sin, Cos, And Tan

One common mistake is mixing up opposite and adjacent. Those labels only make sense after you choose the angle first.

Another common mistake is treating SOHCAHTOA as if it covers every trig problem. It covers the right-triangle definition. If the problem uses general angles, the unit circle is usually the better model.

Students also sometimes forget that tangent is a ratio, not a side length. In a triangle, it compares rise to run.

Another mistake is assuming tangent always exists. In the unit-circle view, $\tan \theta$ is undefined when $\cos \theta = 0$.

Where Sine, Cosine, And Tangent Show Up

They are especially common in:

• right-triangle problems
• slopes and direction
• circular motion and waves
• coordinate geometry and the unit circle

If the problem is about a right triangle, start with the side-ratio view. If it is about angles around a circle, start with the unit-circle view.

Try A Similar Problem

Take the same $3$-$4$-$5$ triangle and switch to the other acute angle. Relabel opposite and adjacent, then recompute $\sin \theta$, $\cos \theta$, and $\tan \theta$. That quick check shows why trig ratios depend on the angle you choose.