Trigonometric Ratios for Special Angles: A Complete Guide
In this guide, we’ll cover the fundamentals of trigonometric ratios and explore the values of trigonometric functions for key special angles like 0°, 30°, 45°, 60°, and 90°. You’ll also learn how to calculate these values and apply them in problems.
What are Trigonometric Ratios?
Trigonometric ratios—sine, cosine, and tangent—are the basis of trigonometry. These ratios relate the sides of a right triangle to its angles:
- Sine (sin θ) = Opposite Side / Hypotenuse
- Cosine (cos θ) = Adjacent Side / Hypotenuse
- Tangent (tan θ) = Opposite Side / Adjacent Side
Each of these ratios has a reciprocal:
- Cosecant (cosec θ) = 1 / sin θ
- Secant (sec θ) = 1 / cos θ
- Cotangent (cot θ) = 1 / tan θ
Special Angles in Trigonometry
Special angles—0°, 30°, 45°, 60°, and 90°—are often used in trigonometry because their trigonometric ratios yield exact values.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
These angles form the basis of trigonometric calculations for exact values, which are frequently tested in math exams and applied in fields like engineering and physics.
Deriving Trigonometric Ratios for 30º, 45º, and 60º
To understand the values of trigonometric ratios for special angles, we can use specific triangles:
1. 30° and 60° Values Using an Equilateral Triangle
In an equilateral triangle split into two right triangles:
- sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
- sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3
2. 45° Values Using an Isosceles Right Triangle
In a right-angled isosceles triangle:
- sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1
Using Special Angles to Solve Problems
Special angle values allow us to solve trigonometric equations and simplify expressions without a calculator. Here’s an example of how to use special angle values:
Example Problem
Find the exact value of sin 105º.
To solve this, use the angle addition formula:
sin (a + b) = sin a * cos b + cos a * sin b
For sin 105°, rewrite as sin (60° + 45°):
- sin 105° = sin 60° * cos 45° + cos 60° * sin 45°
- = (√3/2)(1/√2) + (1/2)(1/√2)
- = √3/2√2 + 1/2√2
Trigonometric Values for 0º, 90º, 180º, 270º, and 360º
These angles are used to complete the unit circle, showing how sine and cosine values fluctuate:
- sin 0º = 0, sin 90º = 1, sin 180º = 0, sin 270º = -1, sin 360º = 0
- cos 0º = 1, cos 90º = 0, cos 180º = -1, cos 270º = 0, cos 360º = 1
- tan 0º = 0, tan 90º = undefined, tan 180º = 0, tan 270º = undefined, tan 360º = 0
These values simplify calculations and help in analyzing trigonometric functions across one complete cycle.
Quick Reference Table of Trigonometric Ratios for Special Angles
For quick access to trigonometric ratios of special angles, here’s a summary:
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Use this table when working with special angles in trigonometry for faster and more accurate calculations.
