Step by Step
1
The 30-60-90 triangle
Side ratios are 1 : √3 : 2 — the shortest leg (opposite the 30° angle) is 1, the longer leg (opposite the 60° angle) is √3, and the hypotenuse is 2.
2
The 45-45-90 triangle
Side ratios are 1 : 1 : √2 — both legs are equal length (1), and the hypotenuse is √2, since this triangle is isosceles.
3
Reading off exact values from 30-60-90
Using SOH-CAH-TOA on the 30-60-90 triangle: sin(30°) = 1/2, cos(30°) = √3/2, sin(60°) = √3/2, cos(60°) = 1/2.
4
Reading off exact values from 45-45-90
Using SOH-CAH-TOA on the 45-45-90 triangle: sin(45°) = cos(45°) = √2/2 (since both legs are equal), and tan(45°) = 1 (since the two legs are equal, their ratio is 1).
Applied Walkthrough
1
Find the exact value of sin(60°) without a calculator, using the 30-60-90 triangle.
2
Recall the side ratios: 1 (short leg, opposite 30°), √3 (long leg, opposite 60°), 2 (hypotenuse).
3
Since 60° is opposite the long leg (√3), and sine is opposite/hypotenuse: sin(60°) = √3/2.
4
As a sanity check, this matches the well-known exact value of sin(60°) = √3/2 ≈ 0.866, confirming the special triangle method works.
Exam Application
Exams test whether you have memorized the side ratios for both special right triangles, and whether you can correctly apply SOH-CAH-TOA to extract exact trig values at 30°, 45°, and 60° without a calculator.
⚠ Common Trap
The most common trap is mixing up which leg goes with which angle in the 30-60-90 triangle — remember the shorter leg is always opposite the smaller angle (30°), and the longer leg (√3) is opposite the larger angle (60°).
✓ Quick Self-Check
1. What are the side ratios of a 30-60-90 triangle?
1 : √3 : 2 (short leg : long leg : hypotenuse).
Tap to reveal / hide
2. What are the side ratios of a 45-45-90 triangle?
1 : 1 : √2.
Tap to reveal / hide
3. What is sin(60°)?
√3/2.
Tap to reveal / hide
4. What is sin(45°)?
√2/2.
Tap to reveal / hide
5. What is tan(45°), and why?
1, because both legs of a 45-45-90 triangle are equal, making their ratio 1.
Tap to reveal / hide