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Mom's Basement University

Introduction to Trigonometry and the Trigonometric Ratios

What You’ll Learn

Section titled “What You’ll Learn”

In this first Trigonometry lesson you’ll learn the three primary trigonometric ratios - sine, cosine, and tangent - and how to use them to relate the angles and sides of right triangles.

The Concept

Section titled “The Concept”

Trigonometry is the study of the relationships between the angles and sides of triangles. Most of the beginning work focuses on right triangles (triangles with one 90° angle).

For any acute angle θ (theta) in a right triangle, we define three important ratios.

(As a side note, θ is a Greek letter that mathematicians use as a standard name for angles. You’ll see it everywhere in trigonometry.)

Sine of θ:

sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

Cosine of θ:

cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}

Tangent of θ:

tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

These are commonly remembered with the mnemonic SOH-CAH-TOA (we covered this in the intro page, but it’s worth repeating because it’s that important):

  • SOH → Sine = Opposite / Hypotenuse
  • CAH → Cosine = Adjacent / Hypotenuse
  • TOA → Tangent = Opposite / Adjacent

The hypotenuse is always the side opposite the right angle (the longest side). The opposite side is the one directly across from angle θ. The adjacent side is the one next to angle θ (not the hypotenuse).

Worked Example

Section titled “Worked Example”

Consider a right triangle where angle θ = 30°, the side opposite θ is 5 units, and the hypotenuse is 10 units.

1. Find sin(30°)

Since we’re finding sine, we use SOH: Opposite / Hypotenuse.

sin⁡(30°)=510=0.5\sin(30°) = \frac{5}{10} = 0.5

2. Find the adjacent side using the Pythagorean Theorem

Adjacent=102−52=100−25=75≈8.66\text{Adjacent} = \sqrt{10^2 - 5^2} = \sqrt{100 - 25} = \sqrt{75} \approx 8.66

3. Find cos(30°) and tan(30°)

cos⁡(30°)≈8.6610=0.866\cos(30°) \approx \frac{8.66}{10} = 0.866 tan⁡(30°)≈58.66≈0.577\tan(30°) \approx \frac{5}{8.66} \approx 0.577

Real-World Application

Section titled “Real-World Application”

Right triangle trigonometry is used in many practical fields:

  • Construction and roofing (calculating rafter lengths and slopes)
  • Navigation and surveying (measuring distances you can’t walk directly)
  • Engineering (designing ramps, bridges, and structures with specific angles)
  • Physics (resolving forces into components)
  • Everyday tasks (finding the height of a tree or building using a simple angle measurement)

Example: a ladder leans against a wall forming a 68° angle with the ground. Using trigonometry, you can calculate how high up the wall the ladder reaches if you know the ladder’s length.

Quiz

Section titled “Quiz”
In a right triangle, $\sin(\theta)$ equals:
The mnemonic SOH-CAH-TOA helps you remember:
If the adjacent side is 12 and the hypotenuse is 15, $\cos(\theta)$ equals:
Which side of a right triangle is always the longest?
If the opposite side is 7 and the adjacent side is 24, $\tan(\theta)$ equals: