Introduction to Trigonometry

Trigonometry is the branch of mathematics that studies relationships between the angles and sides of triangles. The word comes from Greek: trigonon (triangle) + metron (measure). If geometry taught you how to work with shapes, trigonometry gives you the tools to connect angles to distances — and that connection is what makes it so powerful in the real world.

How Trigonometry Connects to Geometry

In geometry, you learned the Pythagorean theorem: $a^2 + b^2 = c^2$. That formula tells you how the three sides of a right triangle relate to each other, but it says nothing about the angles. Trigonometry fills that gap. Given one angle and one side of a right triangle, trig lets you find every other measurement — the remaining sides and the remaining angles.

This works because of a key geometric fact: all right triangles with the same acute angle are similar. That means their sides are always in the same ratio, no matter how large or small the triangle is. Trigonometry gives names to those ratios.

The Right Triangle: Foundation of Trig

Everything in introductory trigonometry starts with a right triangle — a triangle with one 90-degree angle. The other two angles are acute (less than 90 degrees). We typically label one of the acute angles $\theta$ (the Greek letter “theta”) and describe the sides relative to that angle.

The three sides of a right triangle, relative to a chosen angle $\theta$, are:

• Opposite — the side directly across from angle $\theta$
• Adjacent — the side next to angle $\theta$ (that is not the hypotenuse)
• Hypotenuse — the longest side, always across from the 90-degree angle

These labels change depending on which angle you pick. The hypotenuse is always the same, but “opposite” and “adjacent” swap if you switch to the other acute angle.

The Three Basic Trig Ratios (Preview)

Trigonometry defines three fundamental ratios based on the sides of a right triangle:

These three ratios are the building blocks of everything in trigonometry. For any given angle $\theta$, these ratios are constant regardless of the triangle’s size. A 30-degree angle in a small triangle produces the exact same ratios as a 30-degree angle in a large triangle.

You will learn the full details — and the memory trick SOH CAH TOA — in the next topic.

Why Trigonometry Matters

Trigonometry shows up across dozens of professions and academic fields:

• Carpentry and construction — calculating roof pitch, rafter lengths, stair angles, and verifying structural angles
• Electrical work — determining conduit bending angles, analyzing alternating current (AC) circuits, and calculating phase angles
• Surveying and navigation — measuring distances and heights that cannot be reached directly
• Physics and engineering — resolving forces into components, analyzing wave motion, and modeling oscillations
• Standardized tests — trig appears on the SAT, ACT, and GED math sections

Example 1: Roof Pitch

A carpenter needs to build a roof with a 30-degree pitch. The horizontal run from the wall to the peak is 12 feet. Using the tangent ratio:

$\tan(30°) = \frac{\text{rise}}{\text{run}} = \frac{\text{rise}}{12}$

$\text{rise} = 12 \times \tan(30°) \approx 12 \times 0.5774 \approx 6.93 \text{ ft}$

The roof rises about 6.93 feet. Without trigonometry, the carpenter would have to guess or build a scale model.

Example 2: Surveying a Building Height

A surveyor stands 50 meters from a building and measures a 40-degree angle up to the roofline. Using tangent:

$\tan(40°) = \frac{\text{height}}{50}$

$\text{height} = 50 \times \tan(40°) \approx 50 \times 0.839 \approx 41.95 \text{ m}$

The building is approximately 42 meters tall — measured without climbing it.

Example 3: Checking a Miter Cut

A carpenter cuts a board at an angle and wants to verify the cut is exactly 45 degrees. They measure the cut face (hypotenuse) as 5.66 inches and the board thickness (one leg) as 4 inches:

$\sin\theta = \frac{4}{5.66} \approx 0.707$

$\theta = \sin^{-1}(0.707) = 45°$

The cut is confirmed as a perfect 45-degree miter.

What You Will Learn Next

This topic cluster builds your trig skills step by step:

1. SOH CAH TOA — the three basic ratios defined and memorized
2. Finding missing sides — using trig to calculate unknown lengths
3. Finding missing angles — using inverse trig to find unknown angles
4. The unit circle — extending trig beyond right triangles
5. Radians and degrees — the two ways to measure angles
6. Special angles — exact values you need to know by heart
7. Graphs of sine and cosine — how trig functions create waves
8. Trig identities — equations that are always true
9. Law of sines and cosines — solving any triangle

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Trigonometry studies the relationships between angles and side lengths in triangles
• Every right triangle has three sides relative to a chosen angle: opposite, adjacent, and hypotenuse
• The three basic trig ratios — sine, cosine, and tangent — are constant for any given angle regardless of triangle size
• Trig connects angles to distances, which is why it is essential in construction, electrical work, surveying, and standardized tests
• All of trigonometry builds on the right triangle and these three ratios

Return to Trigonometry for more topics in this section.