Trigonometry — Complete Guide

Trigonometry connects angles to lengths. When you need a missing side or angle in a right triangle, it is usually the tool, and the same ideas extend to the unit circle, rotation, and repeating patterns such as waves. The whole subject grows from three ratios.

The Formulas And What Each Side Name Means

For an acute angle $\theta$ in a right triangle,

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

The side names are defined relative to the chosen 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. SOHCAHTOA helps you recall which ratio uses which pair, but only after you label the sides correctly.

Why The Ratios Stay Constant

The formulas look like three separate rules, but they rest on one fact: triangles with the same angles have the same side ratios. If two right triangles share the same acute angles, they are similar, so corresponding sides scale by the same factor and the ratios stay fixed.

That is why $\sin 30^\circ$ or $\cos 60^\circ$ has one fixed value. The triangle can grow or shrink, but as long as the angle stays the same, the ratio does not change. So a trig value depends on the angle, not on the size of the triangle.

Worked Example: Find A Building Height

Stand $20$ meters from a building on level ground, with the angle of elevation to the top equal to $35^\circ$. Ignoring 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 match:

Solve for the height:

Using a calculator in degree mode,

So the building is about $14$ meters tall under those conditions. The general pattern: identify the known side, identify the angle, choose the trig ratio that connects them, and solve.

Now You Try

Use the same setup with a tree: stand $15$ meters away, take an angle of elevation of $40^\circ$, and estimate the height. Choose the ratio before you compute — the adjacent side and the angle are known, so tangent fits again. Set up $\tan 40^\circ = \tfrac{\text{height}}{15}$, then multiply. If you reached for the correct ratio first, you are using the main idea correctly.

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

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

Calculation Traps To Watch

One common trap is labeling opposite and adjacent before choosing the angle. Those labels are relative, not permanent parts of the triangle. Another is using the right ratio for the wrong triangle type: the basic $\sin$, $\cos$, and $\tan$ side-ratio definitions apply directly to right triangles, while non-right triangles usually need the law of sines or the law of cosines. Calculator mode causes errors too — if the problem gives angles in degrees, your calculator must be in degree mode, and radians must match radian work. Finally, $\tan \theta$ 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: 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.