Trigonometry — Complete Guide

Trigonometry is the part of math that connects angles to lengths. If you need to find a missing side or angle in a right triangle, trigonometry is usually the tool. The same ideas also extend to the unit circle, rotation, and repeating patterns such as waves.

Most students start with three functions: sine, cosine, and tangent. For an acute angle $\theta$ in a right triangle,

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

If $\cos \theta \ne 0$, then

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

The key idea is simpler than the formulas: triangles with the same angles have the same side ratios. That is why a trig value depends on the angle, not on the size of the triangle.

What Trigonometry Means In Practice

In a right triangle, trigonometry lets you connect one angle to a pair of side lengths. Once you choose the angle, the side names become relative to that angle.

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

If you switch to a different angle in the same triangle, opposite and adjacent can switch too. That is a common source of mistakes.

Why Sine, Cosine, And Tangent Stay Constant

If two right triangles have the same acute angles, they are similar. Their side lengths may be different, but corresponding sides scale by the same factor. Because of that, the ratios stay the same.

That is why $\sin 30^\circ$ or $\cos 60^\circ$ has one fixed value. The triangle can get larger or smaller, but the ratio does not change as long as the angle stays the same.

Sine, Cosine, And Tangent At A Glance

Each ratio compares a different pair of sides:

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

SOHCAHTOA can help you remember the pattern, but it only helps after you label the sides correctly.

Worked Example: Find A Building Height

Suppose you stand $20$ meters from a building on level ground and the angle of elevation to the top is $35^\circ$. If you ignore eye height, how tall is the building?

This is a right-triangle problem. The horizontal distance is the adjacent side, and the building height is the opposite side. Since we know the angle and the adjacent side, tangent is the best match:

$$\tan 35^\circ = \frac{\text{height}}{20}$$

Solve for the height:

$$\text{height} = 20 \tan 35^\circ$$

Using a calculator in degree mode,

$$\text{height} \approx 20(0.7002) \approx 14.0$$

So the building is about $14$ meters tall under those conditions.

The general pattern is simple: identify the known side, identify the angle, choose the trig ratio that connects them, and solve.

Where The Unit Circle Fits

Right triangles are the starting point, not the whole story. To work with angles larger than $90^\circ$, negative angles, or full rotations, trigonometry extends to the unit circle.

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

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

So cosine is the horizontal coordinate and sine is the vertical coordinate. This is why the same functions also describe circular motion and periodic graphs.

Common Trigonometry Mistakes

One common mistake is labeling opposite and adjacent before choosing the angle. Those labels are relative, not permanent parts of the triangle.

Another mistake is using the right ratio for the wrong triangle type. The basic $\sin$, $\cos$, and $\tan$ side-ratio definitions apply directly to right triangles. For non-right triangles, you usually need tools such as the law of sines or the law of cosines.

Calculator mode also causes errors. If the problem gives angles in degrees, your calculator must be in degree mode. If the work is in radians, the calculator must match that.

It also helps to remember that $\tan \theta$ is not defined when $\cos \theta = 0$, because division by zero is not allowed.

When Trigonometry Is Used

Trigonometry appears whenever direction, rotation, height, distance, or periodic change matters. Common examples include surveying, navigation, engineering, physics, computer graphics, and signal analysis.

In school math, you will usually meet it in four forms: right-triangle problems, unit-circle values, trigonometric identities, and graphs of sine and cosine.

Try A Similar Problem

Try the same setup with a tree instead of a building: stand $15$ meters away, use an angle of elevation of $40^\circ$, and estimate the height. If you can choose the correct ratio before calculating, you are using the main idea correctly.