Sin, Cos, Tan — Explained

Sine, cosine, and tangent are three ratios that compare side lengths relative to one chosen acute angle in a right triangle. Once you fix which side is opposite, adjacent, and hypotenuse, all three become straightforward to compute.

The Three Ratios And Their Symbols

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

The sides are defined relative to $\theta$:

• 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.

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.

Why These Ratios Are Fixed By The Angle

For a fixed angle, these ratios stay the same even if the triangle is scaled up or down, because similar triangles keep the same side ratios. That is what makes sine, cosine, and tangent functions of the angle alone.

It also explains a key fact: 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.

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

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.

Practice: Switch To The Other Angle

Keep the same $3$-$4$-$5$ triangle, but now let $\theta$ be the acute angle opposite the side of length $4$. Relabel before computing: opposite $= 4$, adjacent $= 3$, hypotenuse $= 5$.

Check your work against these values:

The numbers in the triangle did not change, but choosing a different angle swapped opposite and adjacent. That is the whole reason trig ratios depend on the angle you pick.

When To Use Sine, Cosine, Or Tangent

• 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.

These ratios are especially common in right-triangle problems, slopes and direction, circular motion and waves, and coordinate geometry.

How The Unit Circle Extends The Same Idea

The right-triangle definitions apply directly to acute angles. For angles larger than $90^\circ$, negative angles, or full rotations, the same functions extend using the unit circle, where the point for angle $\theta$ is

so cosine is the $x$-coordinate and sine is the $y$-coordinate, and tangent is still

when $\cos \theta \ne 0$. This is why the same trig names keep working even when there is no right triangle drawn.

Pitfalls To Watch For

The most frequent slip is mixing up opposite and adjacent. Those labels only make sense after you choose the angle first.

A second is treating SOHCAHTOA as if it covers every trig problem. It covers the right-triangle definition; for general angles, the unit circle is the better model.

Students also forget that tangent is a ratio, not a side length, and that in the unit-circle view $\tan \theta$ is undefined when $\cos \theta = 0$.

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.