📐 Trigonometry Calculator
Calculate trigonometric functions, solve triangles, and explore the unit circle
Trigonometric Functions
Inverse Functions & Conversion
📖 How to Use the Trigonometry Calculator
Step-by-Step Guide:
- Input Method: Enter your angle value in the input field
- Select Unit: Choose between degrees or radians using the dropdown
- Choose Function: Click the desired trigonometric function button (sin, cos, tan, etc.)
- View Results: The calculator displays the result with high precision
- Clear & Repeat: Use the clear button to reset and perform new calculations
Best Practices:
- Use parentheses for complex expressions
- Remember π ≈ 3.14159 for radian calculations
- Check your angle unit (degrees vs radians)
- For inverse functions, ensure input is within valid domain
- Verify results using trigonometric identities
🔗 Related Calculators
🌍 Real-Life Applications
🏗️ Civil Engineering
Bridge Construction: Engineers use trigonometry to calculate the angles and forces in bridge supports and trusses.
Example: To find the height of a building from a distance of 50m at a 30° angle of elevation: height = 50 × tan(30°) ≈ 28.87m
Road Design: Calculating slopes, banking angles, and curve radii for safe vehicle navigation.
🌊 Physics & Wave Analysis
Sound Waves: Audio engineers use sine and cosine functions to analyze and manipulate sound frequencies.
Example: A sound wave can be represented as y = A × sin(2πft), where A is amplitude, f is frequency, and t is time.
Electromagnetic Waves: Radio, WiFi, and cellular signals all follow trigonometric wave patterns.
🧭 Navigation & GPS
Satellite Positioning: GPS systems use trigonometry to triangulate your exact position using signals from multiple satellites.
Example: Ships use the bearing formula: bearing = atan2(sin(Δlong)cos(lat2), cos(lat1)sin(lat2) - sin(lat1)cos(lat2)cos(Δlong))
Aviation: Pilots calculate flight paths, wind compensation, and landing approaches.
🎮 Computer Graphics & Gaming
3D Rendering: Game engines use trigonometry for rotating objects, calculating lighting angles, and camera movements.
Example: Rotating a point (x,y) by angle θ: x' = x×cos(θ) - y×sin(θ), y' = x×sin(θ) + y×cos(θ)
Animation: Smooth character movements and realistic physics simulations rely on trigonometric functions.
🏥 Medical Imaging
CT Scans: Computed tomography uses trigonometric reconstruction algorithms to create 3D images from 2D X-ray slices.
Example: The Radon transform uses trigonometry to convert projection data into cross-sectional images.
Ultrasound: Doppler effect calculations use trigonometry to measure blood flow and heart function.
🌟 Astronomy & Space
Orbital Mechanics: Calculating satellite orbits, spacecraft trajectories, and planetary positions.
Example: The period of a satellite orbit: T = 2π√(r³/GM), where trigonometry helps determine orbital positions.
Telescope Positioning: Astronomers use trigonometry to track celestial objects and coordinate observations.
Related Math Tools
❓ Frequently Asked Questions
Degrees: A circle is divided into 360 degrees. Common in everyday use and navigation.
Radians: Based on the radius of a circle. One full circle = 2π radians ≈ 6.28 radians.
Conversion: Degrees × (π/180) = Radians | Radians × (180/π) = Degrees
Example: 90° = π/2 radians ≈ 1.57 radians
Division by Zero: tan(90°) and sec(90°) are undefined because they involve division by cos(90°) = 0.
Domain Restrictions: Inverse functions have limited domains:
- arcsin(x) and arccos(x): x must be between -1 and 1
- arctan(x): defined for all real numbers
Common Undefined Values: tan(90°), tan(270°), cot(0°), cot(180°), sec(90°), csc(0°)
Precision: The calculator uses JavaScript's built-in Math functions with double-precision floating-point accuracy (about 15-17 decimal digits).
Rounding: Results are displayed with reasonable precision to avoid showing unnecessary decimal places.
Scientific Applications: Suitable for most engineering, scientific, and educational purposes.
Limitations: Very large or very small numbers may experience floating-point precision limits.
Fundamental Identity: sin²(θ) + cos²(θ) = 1
Reciprocal Identities:
- csc(θ) = 1/sin(θ)
- sec(θ) = 1/cos(θ)
- cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)
Pythagorean Identities:
- 1 + tan²(θ) = sec²(θ)
- 1 + cot²(θ) = csc²(θ)
Sum/Difference Formulas: sin(A ± B) = sin(A)cos(B) ± cos(A)sin(B)
Step 1: Isolate the trigonometric function (sin, cos, or tan)
Step 2: Use inverse functions to find the reference angle
Step 3: Consider all possible solutions within the given domain
Example: Solve sin(x) = 0.5
- x = arcsin(0.5) = 30° or π/6 radians
- General solutions: x = 30° + 360°n or x = 150° + 360°n (where n is any integer)
Tip: Remember that sine and cosine have period 2π, while tangent has period π
Engineering Fields:
- Civil Engineering: Building design, surveying, structural analysis
- Electrical Engineering: AC circuits, signal processing, wave analysis
- Mechanical Engineering: Vibration analysis, robotics, fluid dynamics
- Aerospace Engineering: Flight dynamics, orbital mechanics
Technology & Science:
- Computer Graphics: Game development, 3D modeling, animation
- Physics: Wave mechanics, optics, quantum mechanics
- Astronomy: Celestial navigation, telescope positioning
- Medical Imaging: CT scans, MRI, ultrasound technology
Other Applications: Architecture, Navigation, Surveying, Music & Audio Engineering
📚 Trigonometry Reference
Basic Trigonometric Functions
Common Angles
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° (0) | 0 | 1 | 0 |
| 30° (π/6) | 1/2 | √3/2 | √3/3 |
| 45° (π/4) | √2/2 | √2/2 | 1 |
| 60° (π/3) | √3/2 | 1/2 | √3 |
| 90° (π/2) | 1 | 0 | ∞ |
Trigonometric Identities
Pythagorean Identity:
Double Angle Formulas:
Applications
Engineering & Physics:
- Wave analysis and signal processing
- Oscillatory motion and pendulums
- AC circuit analysis
- Structural engineering calculations
Navigation & Astronomy:
- GPS and satellite positioning
- Celestial navigation
- Distance and height calculations