📐 TRIGONOMETRY FUNCTIONS
Angles, Radians, Degrees & Trigonometric Identities
Angle Definition
Measure of rotation of a ray about its initial point
Initial Side
The original ray before rotation
Terminal Side
Final position of the ray after rotation
Positive Angle
Rotation is anticlockwise (counterclockwise)
📏 Angle Measurement Formula
\[ \theta = \frac{l}{n} \]
\( l \)
One degree (1°)
\( n \)
One minute (1′)
1″
One second
📝 Note on Units
1 degree (1°) = 60 minutes
1 minute (1′) = 60 seconds
1° = 3600 seconds
🔄 Conversion Tables
Radian → Degree
| Radian Measure | Degree Measure |
|---|---|
| \[ \frac{\pi}{180} \] | \( x \) |
Formula: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \)
Degree → Radian
| Degree Measure | Radian Measure |
|---|---|
| \[ \frac{180}{\pi} \] | \( x \) |
Formula: \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \)
Even & Odd Trigonometric Functions
| Function | Property | Type |
|---|---|---|
| \( \sin(-x) \) | \( = -\sin(x) \) | Odd Function |
| \( \cos(-x) \) | \( = \cos(x) \) | Even Function |
| \( \tan(-x) \) | \( = -\tan(x) \) | Odd Function |
| \( \cot(-x) \) | \( = -\cot(x) \) | Odd Function |
📊 Angle Values & Relationships
Common Angle Relationships
\[ \theta = \frac{\pi}{180} \]
Conversion factor
\[ \theta = \frac{\pi}{180} \text{ radian} \]
Angle in radians
\[ 1^\circ = \frac{\pi}{180} \text{ radians} \]
Degree to radian
\[ 1 \text{ radian} = \frac{180}{\pi} \text{ degrees} \]
Radian to degree
🎯 Key Relationships
\( \pi \) radians = 180°
\( 2\pi \) radians = 360°
\( \frac{\pi}{2} \) radians = 90°
\( \frac{\pi}{3} \) radians = 60°
\( \frac{\pi}{4} \) radians = 45°
\( \frac{\pi}{6} \) radians = 30°
🔢 Fundamental Trigonometric Identities
Pythagorean Identities
\[ \sin^2 x + \cos^2 x = 1 \]
Basic Pythagorean Identity
\[ 1 + \tan^2 x = \sec^2 x \]
Derived from dividing by \( \cos^2 x \)
\[ 1 + \cot^2 x = \csc^2 x \]
Derived from dividing by \( \sin^2 x \)
📝 Note: Alternative Notation
\( \csc x \) = \( \frac{1}{\sin x} \)
(Cosecant)
\( \sec x \) = \( \frac{1}{\cos x} \)
(Secant)
\( \cot x \) = \( \frac{1}{\tan x} \)
(Cotangent)