Trigonometry Calculator
Calculate trigonometric functions with detailed explanations and visualizations
Trigonometric Functions
Trigonometric Results
Enter an angle to calculate the trigonometric function
Understanding Trigonometry
What is Trigonometry?
Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. The word comes from Greek "trigonon" meaning "triangle" and "metron" meaning "measure".
Basic Trigonometric Ratios:
sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent
These ratios are defined for right-angled triangles
Unit Circle
The unit circle is a circle with radius 1 centered at the origin. It's fundamental to understanding trigonometric functions as they relate angles to coordinates on this circle.
- •Coordinates: (cos θ, sin θ) for angle θ
- •Periodicity: Functions repeat every 360° or 2π radians
- •Quadrants: Signs of functions change in different quadrants
- •Radians: Natural unit for angles in calculus
Special Angles
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0.0000 | 0 | 1 | 0 |
| 30° | 0.5236 | 0.5 | 0.866025 | 0.57735 |
| 45° | 0.7854 | 0.707107 | 0.707107 | 1 |
| 60° | 1.0472 | 0.866025 | 0.5 | 1.732051 |
| 90° | 1.5708 | 1 | 0 | ∞ |
| 180° | 3.1416 | 0 | -1 | 0 |
| 270° | 4.7124 | -1 | 0 | ∞ |
| 360° | 6.2832 | 0 | 1 | 0 |
Real-World Applications
Physics
Motion and forces
- • Projectile motion
- • Wave motion
- • Oscillations
- • Circular motion
Engineering
Design and analysis
- • Structural analysis
- • Signal processing
- • Control systems
- • Mechanical design
Navigation
Position and direction
- • GPS coordinates
- • Bearing calculations
- • Course plotting
- • Astronomy
Computer Graphics
3D rendering
- • 3D rotations
- • Camera projections
- • Animation
- • Game physics
Architecture
Building design
- • Roof slopes
- • Structural angles
- • Surveying
- • Construction
Music
Sound waves
- • Sound synthesis
- • Acoustics
- • Harmonics
- • Audio engineering
Trigonometric Identities
Fundamental Identities
Pythagorean Identity
sin²θ + cos²θ = 1
Reciprocal Identities
cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ
Quotient Identities
tanθ = sinθ/cosθ, cotθ = cosθ/sinθ
Angle Sum and Difference
Sum Formulas
sin(α±β) = sinα cosβ ± cosα sinβ
cos(α±β) = cosα cosβ ∓ sinα sinβ
Product Formulas
sinα sinβ = ½[cos(α-β) - cos(α+β)]
cosα cosβ = ½[cos(α-β) + cos(α+β)]
Double Angle Formulas
sin(2θ) = 2sinθ cosθ
Double angle for sine
cos(2θ) = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ
Double angle for cosine
tan(2θ) = 2tanθ/(1 - tan²θ)
Double angle for tangent
Inverse Trigonometric Functions
Definition and Domain
Inverse trigonometric functions find the angle when the trigonometric ratio is known. They are denoted with arc- prefix.
arcsin(x) = sin⁻¹(x)
Domain: [-1, 1], Range: [-π/2, π/2]
arccos(x) = cos⁻¹(x)
Domain: [-1, 1], Range: [0, π]
arctan(x) = tan⁻¹(x)
Domain: (-∞, ∞), Range: (-π/2, π/2)
Common Values
arcsin(1/2) = π/6 (30°)
arcsin(√2/2) = π/4 (45°)
arccos(1/2) = π/3 (60°)
arccos(√2/2) = π/4 (45°)
arctan(1) = π/4 (45°)
arctan(√3) = π/3 (60°)