Lesson 8.2.3

Intro to Trigonometry (SOH CAH TOA)

Trigonometry connects angles and sides of right triangles through three ratios: sine, cosine, and tangent. The mnemonic SOH CAH TOA is your key to remembering them.

Introduction

Until now, we've needed two sides to find the third (Pythagorean Theorem) or two angles to prove similarity. Trigonometry changes everything: with just one side and one acute angle, you can find every other measurement in a right triangle.

Past Knowledge

Pythagorean Theorem (8.1). Special right triangles (8.2.1–8.2.2). AA Similarity.

Today's Goal

Define sin, cos, and tan; identify opposite, adjacent, and hypotenuse relative to an angle.

Future Success

Tangent (8.2.4), Sine & Cosine (8.2.5), inverse trig (8.3.1), Law of Sines/Cosines.

Key Concepts

SOH CAH TOA

Sine = Opposite / Hypotenuse:

Cosine = Adjacent / Hypotenuse:

Tangent = Opposite / Adjacent:

Identifying the Sides

Hypotenuse: Always the longest side, opposite the 90° angle
Opposite: The side across from the angle you're working with
Adjacent: The side next to the angle (not the hypotenuse)

Why This Works

By AA Similarity, all right triangles with the same acute angle are similar. That means the ratios of their sides are always the same — those fixed ratios are what we call sine, cosine, and tangent.

Worked Examples

Basic

Writing Trig Ratios

Right triangle with sides 3, 4, 5. Find sin, cos, and tan of the angle opposite the side of length 3.

Opp = 3, Adj = 4, Hyp = 5

, ,

Intermediate

Verifying with Special Triangles

Find , , and using a 45-45-90 triangle.

Sides:

Advanced

30° and 60° Ratios

Find all six trig ratios for 30° and 60° using a 30-60-90 triangle with sides .

For 30°: opp = 1, adj = , hyp = 2

, ,

For 60°: opp = , adj = 1, hyp = 2

, ,

Notice: and — complementary angles swap sin and cos!

Common Pitfalls

Swapping Opposite and Adjacent

Opposite and adjacent change depending on which angle you're looking at. Always identify the angle first, then label sides relative to it.

Using Trig on Non-Right Triangles

SOH CAH TOA only works on right triangles. For non-right triangles, you need the Law of Sines or Law of Cosines (8.3).

Real-Life Applications

Engineering — Force Decomposition

Engineers break forces into horizontal and vertical components using sine and cosine. A 100 N force at 30° contributes N horizontally and N vertically.

Video Games — Physics Engines

Every projectile trajectory, slope movement, and camera angle in a video game uses sin, cos, and tan to calculate positions and velocities in real time.