Lesson 9.4

Special Right Triangles

Before confronting the infinite complexity of the circle, we master the two specific triangles that build it.

Introduction

In Lesson 9.3, we learned SOH CAH TOA for generic triangles. However, 90% of trigonometry revolves around just two very specific triangles: the 30-60-90 and the 45-45-90. Mastering these is the secret to memorizing the unit circle effortlessly.

Prerequisite Connection

This comes directly from Geometry (Isosceles Triangles and Equilateral Triangles split in half).

Today's Increment

We memorize the side ratios: for 45-45-90 and for 30-60-90.

Why This Matters

In Calculus, you will evaluate limits like and compute definite integrals. You will not have a calculator. You must be able to visualize these triangles instantly to find exact values.

Key Concepts

45-45-90 (Isosceles Right)

From a square cut in half diagonally.

Legs: x, x

Hypotenuse:

"Legs are same, Hypotenuse gets root 2"

30-60-90 (Scalene Right)

From an equilateral triangle cut in half.

Short Leg (opp 30): x

Long Leg (opp 60):

Hypotenuse: 2x

"Hypotenuse is double the short leg"

Worked Examples

Example 1: Finding Hypotenuse

Given a 45-45-90 triangle with legs of length 5, find the hypotenuse.

Identify Pattern

Legs are . Hypotenuse is .

Substitute

Hypotenuse:

Example 2: Working Backwards

The hypotenuse of a 30-60-90 triangle is 10. Find the legs.

Find Short Leg First

This is crucial. Always find the short leg (opposite 30) first.
.

Find Long Leg

Long Leg is .

Short Leg: 5
Long Leg:

Example 3: Area Calculation

Find the area of an equilateral triangle with side length 8.

Split into Right Triangles

Drop a height altitude. This splits the base (8) into two segments of 4. Now we have a 30-60-90 triangle.

Find Height

Short leg = 4. Angle opposite height is 60.
Height = Long Leg = .

Calculate Area

Common Pitfalls

Mixing up the 60 and 30 sides

Remember: The Shortest side is always opposite the Smallest angle (30). The Longest side (hypotenuse) is opposite the Largest angle (90).

Real-World Application

Construction & Carpentry

Roof pitches often come in standard ratios. A "12/12 pitch" roof rises 12 inches for every 12 inches of run. This creates a perfect 45-45-90 triangle. Carpenters use this fact ('Hip Rafters' = run x 1.414, where 1.414 is ) to cut beams without measuring every single one.