Related Calculators

Inverse Trigonometry Calculator
Average Calculator
Circle Measurements Calculator
Combinatorics Calculator
BODMAS Calculator
Quadratic Formula Calculator

Inverse Trigonometry Calculator, Average Calculator, Circle Measurements Calculator, Combinatorics Calculator, BODMAS Calculator, Quadratic Formula Calculator

Trigonometric Function Calculator

Calculate sine, cosine, tangent, and other trigonometric functions for any angle. Convert between degrees and radians and visualize results on the unit circle.

Enter the angle value for calculation
Choose whether your angle is in degrees or radians
Select the trigonometric function to calculate
Number of decimal places in the result

How to Use This Calculator

  1. Enter the angle value you want to calculate
  2. Select whether the angle is in degrees or radians
  3. Choose the trigonometric function you want to calculate
  4. Adjust the precision slider to set the number of decimal places
  5. Optionally, select to show the unit circle visualization
  6. Click "Calculate" to see the results

Formula Used

sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent = sin(θ)/cos(θ)
csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)

Where:

  • θ = The angle in degrees or radians
  • opposite = The side opposite to the angle in a right triangle
  • adjacent = The side adjacent to the angle in a right triangle
  • hypotenuse = The longest side of a right triangle

Example Calculation

Real-World Scenario:

Calculate the sine, cosine, and tangent of a 45° angle, which is commonly used in engineering and physics problems.

Given:

  • Angle = 45°
  • Function = sin, cos, tan

Calculation:

First, convert 45° to radians: 45° × π/180 = π/4 radians

sin(45°) = sin(π/4) ≈ 0.7071

cos(45°) = cos(π/4) ≈ 0.7071

tan(45°) = tan(π/4) = 1

Result: For a 45° angle, sin ≈ 0.7071, cos ≈ 0.7071, and tan = 1. This makes sense because in a 45-45-90 triangle, the two legs are equal in length.

Why This Calculation Matters

Practical Applications

  • Engineering and construction for calculating forces and angles
  • Navigation and GPS for determining positions and distances
  • Physics for analyzing wave motion and oscillations
  • Computer graphics for 3D modeling and animations

Key Benefits

  • Solves complex geometric problems quickly
  • Helps visualize relationships between angles and distances
  • Essential for understanding periodic phenomena
  • Foundation for advanced mathematics like calculus

Common Mistakes & Tips

One of the most common mistakes is mixing degrees and radians in calculations. Always ensure your calculator is set to the correct mode. To convert between them: degrees × π/180 = radians, and radians × 180/π = degrees. Our calculator handles this conversion automatically when you select the appropriate unit.

Some trigonometric functions are undefined for certain angles. For example, tan(90°) is undefined because cos(90°) = 0, and tan = sin/cos. Similarly, sec(90°) and csc(0°) are undefined. When calculating these functions, pay attention to these special cases to avoid errors in your calculations.

When using inverse trigonometric functions (arcsin, arccos, arctan), remember that they return principal values in specific ranges. arcsin and arctan return values in [-π/2, π/2], while arccos returns values in [0, π]. This means that for some angles, you may need to add π or 2π to get the correct solution in your specific context.

Frequently Asked Questions

Degrees and radians are two different units for measuring angles. Degrees divide a full circle into 360 equal parts, while radians divide a full circle into 2π parts. One radian is approximately 57.2958 degrees. Radians are often preferred in advanced mathematics and physics because they simplify many formulas, particularly in calculus.

Some trigonometric functions are undefined for certain angles because they involve division by zero. For example, tan(θ) = sin(θ)/cos(θ), so tan(90°) is undefined because cos(90°) = 0. Similarly, sec(θ) = 1/cos(θ) is undefined when cos(θ) = 0, and csc(θ) = 1/sin(θ) is undefined when sin(θ) = 0. These undefined points correspond to vertical asymptotes in the graphs of these functions.

Trigonometric functions have numerous real-world applications. In engineering, they're used to calculate forces, stresses, and angles in structures. In navigation, they help determine positions and distances using triangulation. In physics, they describe wave motion, oscillations, and circular motion. In computer graphics, they're essential for 3D transformations, rotations, and animations. They're also used in signal processing, electrical engineering, astronomy, and many other fields.

References & Disclaimer

Mathematical Disclaimer

This Trigonometric Function Calculator provides results based on standard mathematical formulas. While we strive for accuracy, rounding errors may occur, especially for very large or very small angles. For critical applications, please verify results using multiple sources or consult with a qualified mathematician.

References

Accuracy Notice

This calculator provides results with precision up to 10 decimal places. For angles near points where functions are undefined (like tan(90°)), results may show very large numbers due to the limitations of floating-point arithmetic. In such cases, consult mathematical references for the exact behavior of these functions at these points.

About the Author

Kumaravel Madhavan

Web developer and data researcher creating accurate, easy-to-use calculators across health, finance, education, and construction and more. Works with subject-matter experts to ensure formulas meet trusted standards like WHO, NIH, and ISO.

Connect with LinkedIn