Lesson 9.3

Right Triangle Ratios (SOH CAH TOA)

Before circles, there were triangles. Understanding the 6 fundamental ratios that govern clear geometric relationships.

Introduction

Trigonometry literally means "triangle measurement". The core insight is that for any right triangle with a given acute angle , the ratio of its sides is constant, regardless of how big the triangle is (due to similarity).

Prerequisite Connection

This follows from Similar Triangles (Geometry). If angles match, side ratios match.

Today's Increment

We formalize these ratios as Sine, Cosine, and Tangent (and their reciprocals). The famous mnemonic is SOH CAH TOA.

Why This Matters

In Calculus (and physics), we often decompose forces or movement into horizontal and vertical components. extracts the horizontal part; extracts the vertical part.

Key Concepts

SOH CAH TOA

For a right triangle with angle :

SOH

Sine = Opposite / Hypotenuse

CAH

Cosine = Adjacent / Hypotenuse

TOA

Tangent = Opposite / Adjacent

Reciprocal Functions

Cosecant:

Secant:

Cotangent:

Tip: "S" goes with "C" (Sine/Cosecant, Cosine/Secant).

Worked Examples

Example 1: Finding Ratios

Given a right triangle with legs 3 and 4, find where is opposite the side of length 3.

Find Hypotenuse

Use Pythagorean Theorem: .

Apply SOH

Sine = Opposite / Hypotenuse.

Example 2: Solving for a Side

A ladder leans against a wall making a angle with the ground. If the ladder is 12 ft long, how high up the wall does it reach?

Identify Sides

Angle: .
Hypotenuse: 12 (the ladder).
We want the Height (Opposite to the angle).

Choose Ratio

Relative to , we have Opp and Hyp. That is SOH.

Solve

ft.

Example 3: Inverse Trig

Find if in a right triangle.

Interpret Tangent

. If the ratio is 1, then Opposite = Adjacent.

Identify Triangle

A right triangle with equal legs is an Isosceles Right Triangle.

Common Pitfalls

Relative Perspectives

"Opposite" and "Adjacent" depend on which angle you are standing at. The side "Opposite" to angle A is "Adjacent" to angle B. Always double check which angle is .

Real-World Application

GPS and Surveying

Modern surveying (and GPS triangulation) relies entirely on these ratios. If you know the distance to a satellite (hypotenuse) and the angle of elevation, you can calculate the precise altitude coordinates (Opposite side) using Sine.