Special Right Triangles (30-60-90)
The second “magic” triangle supplies the rest of the unit circle coordinates. Together with the 45-45-90, these two triangles let you evaluate trig functions at every standard angle without a calculator.
Introduction
The 30-60-90 triangle comes from splitting an equilateral triangle in half. This gives us the exact trig values for () and () — the two most tested angles on any exam.
Past Knowledge
The 45-45-90 triangle ratio is . An equilateral triangle has all angles.
Today's Goal
Derive the ratio and obtain the unit circle coordinates for and .
Future Success
Between the 45-45-90 and 30-60-90, you have every tool needed to fill in the entire unit circle.
Key Concepts
Deriving the Ratio from an Equilateral Triangle
Start with an equilateral triangle where every side is . Drop a perpendicular from the top vertex to the base. This altitude:
- Bisects the base into two segments of length
- Creates two congruent 30-60-90 triangles
Find the altitude using the Pythagorean Theorem:
30-60-90 Side Ratio
Short leg is opposite , long leg is opposite
Visualizing the 30-60-90 Triangle
Connecting to the Unit Circle
To place this triangle on the unit circle, the hypotenuse must equal . Divide every side by :
At ()
,
At ()
,
Complete Trig Values
| Function | 30° (π/6) | 60° (π/3) |
|---|---|---|
| sin | ||
| cos | ||
| tan |
Worked Examples
Finding Missing Sides
Question: A 30-60-90 triangle has a hypotenuse of . Find the short and long legs.
Step 1: Short leg = hypotenuse ÷ 2.
Step 2: Long leg = short leg × .
Final Answer: Short leg = , Long leg =
Given the Long Leg
Question: The long leg of a 30-60-90 triangle is . Find the hypotenuse.
Step 1: Find the short leg. Long leg = short leg × , so:
Step 2: Find the hypotenuse. Hypotenuse = short leg × 2.
Final Answer: Hypotenuse
Exact Trig Expression
Question: Find the exact value of .
Step 1: Recall the values.
Step 2: Compute.
Final Answer:
Common Pitfalls
Swapping sin 30° and sin 60°
Students frequently mix up and . Remember: the smaller angle has the smaller sine value. , so .
Confusing 1:√3:2 with 1:1:√2
The 45-45-90 has on the hypotenuse, and the 30-60-90 has on the hypotenuse with on the long leg. Keep them straight by remembering: the 30-60-90 has the simplest hypotenuse (just ).
Real-Life Applications
Hexagonal Structures
A regular hexagon is composed of six equilateral triangles, each containing two 30-60-90 triangles. Honeycomb structures, hex bolts, and even graphene molecular lattices all rely on this geometry. Engineers computing the area or cross-sectional stress of hexagonal shapes use the ratio constantly.
Practice Quiz
Loading...