Trigonometric Functions Calculator

Calculate all six trigonometric functions for a given angle in degrees or radians.

Enter an angle and select its unit (degrees or radians) to compute the values of sine, cosine, tangent, cosecant, secant, and cotangent. This tool is perfect for students, engineers, and anyone working with trigonometry.

Practical Examples

Explore these examples to see how to use the trigonometric functions calculator for common angles and scenarios.

Sine of 30 Degrees

findSin

Calculating the sine of a 30-degree angle, a common value in right-angled triangles.

Angle: 30

Unit: degrees

Cosine of π/4 Radians

findCos

Calculating the cosine for an angle of π/4 radians (equivalent to 45 degrees).

Angle: 0.785398

Unit: radians

Tangent of 90 Degrees

findTan

Illustrating a case where the tangent function is undefined, which occurs at 90 degrees.

Angle: 90

Unit: degrees

Cosecant of π/2 Radians

findCsc

Calculating the cosecant (reciprocal of sine) for an angle of π/2 radians (90 degrees).

Angle: 1.570796

Unit: radians

Other Titles
Understanding Trigonometric Functions: A Comprehensive Guide
An in-depth look at the six trigonometric functions, their properties, applications, and the mathematics behind them. This guide will help you master trigonometry from basic concepts to advanced applications.

What Are Trigonometric Functions?

  • The Core Concepts of Trigonometry
  • Right-Angled Triangles and SOHCAHTOA
  • The Unit Circle and Function Definitions
Trigonometric functions are fundamental mathematical functions that establish a relationship between the angles and side lengths of a right-angled triangle. The six primary functions are sine (sin), cosine (cos), tangent (tan), and their reciprocals: cosecant (csc), secant (sec), and cotangent (cot). They are essential across various fields, including physics, engineering, computer graphics, and navigation.
SOHCAHTOA: A Mnemonic for Right Triangles
For an angle θ in a right-angled triangle, the primary functions are defined by the ratios of the lengths of its sides: the Opposite side, the Adjacent side, and the Hypotenuse. The mnemonic SOHCAHTOA is a helpful way to remember these ratios: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.
The Unit Circle
The unit circle (a circle with a radius of 1) provides a more general definition for trigonometric functions beyond right triangles. For a point (x, y) on the unit circle corresponding to an angle θ, the definitions are: sin(θ) = y, cos(θ) = x, and tan(θ) = y/x. This approach allows for the calculation of trigonometric values for any angle, including those greater than 90° or less than 0°.

Fundamental Ratio Examples

  • For a 30-60-90 triangle, sin(30°) = 1/2
  • On the unit circle at θ = 45° (π/4 rad), the point is (√2/2, √2/2), so cos(45°) = √2/2

Step-by-Step Guide to Using the Trigonometric Functions Calculator

  • Entering the Angle Value
  • Selecting the Correct Angle Unit
  • Interpreting the Calculated Results
Our calculator simplifies the process of finding trigonometric values. Follow these steps to get accurate results quickly and efficiently.
Step 1: Input the Angle
In the 'Angle' input field, enter the numerical value of the angle you wish to evaluate. The calculator accepts both positive and negative values.
Step 2: Choose Degrees or Radians
It is critical to select the correct unit for your angle. Use the 'Angle Unit' selector to choose between 'Degrees' and 'Radians'. This choice significantly affects the result, as sin(90) in degrees is 1, while sin(90) in radians is approximately 0.89.
Step 3: Select the Function and Calculate
Choose the desired function (e.g., Sine, Cosine, Tangent) from the dropdown menu. Click the 'Calculate' button to see the result. The 'Reset' button will clear all fields for a new calculation.

Calculation Examples

  • Input: Angle=45, Unit=Degrees, Function=Tangent -> Result: 1
  • Input: Angle=1.5708, Unit=Radians, Function=Sine -> Result: ≈1 (Approximation for π/2)

Real-World Applications of Trigonometric Functions

  • Physics and Engineering
  • Computer Graphics and Game Development
  • Navigation and Astronomy
Trigonometry is not just an academic subject; it's a practical tool used to solve real-world problems. Its applications are vast and varied.
Physics: Analyzing Waves and Oscillations
Sine and cosine functions are the building blocks for describing periodic phenomena. They model sound waves, light waves, simple harmonic motion (like a pendulum), and AC electrical circuits. The shape of a sine wave perfectly captures the nature of these oscillations.
Computer Graphics: Rendering 3D Worlds
In video games and computer-generated imagery (CGI), trigonometry is used to calculate object rotations, perspective, and lighting. When a character in a game turns, their new position and orientation are computed using trigonometric functions.
Navigation: Finding Your Way
GPS systems, aviation, and astronomy rely on trigonometry to determine locations and distances. By using triangulation and trigonometric principles, we can accurately pinpoint positions on Earth or map the distances between celestial bodies.

Application Use Cases

  • Modeling an AC voltage signal: V(t) = V_max * sin(ωt + φ)
  • Calculating the height of a building by measuring the angle of elevation from a distance.

Common Misconceptions and Correct Methods

  • Mistaking Degrees for Radians
  • Misunderstanding Undefined Values
  • Errors in Reciprocal Function Calculations
Avoiding common errors is key to mastering trigonometry. Understanding these pitfalls will ensure your calculations are always accurate.
The Critical Importance of Angle Units
The most frequent mistake is using the wrong angle unit. Mathematical formulas and programming libraries often default to radians. Always double-check which unit is required. Remember the conversion: 180° = π radians.
Why is Tangent of 90° Undefined?
tan(θ) = sin(θ) / cos(θ). At θ = 90°, cos(90°) = 0. Division by zero is mathematically undefined. Therefore, tan(90°) is undefined. Similarly, cot(0°) is undefined because sin(0°) = 0. The calculator will correctly show 'Undefined' for these cases.
Reciprocal vs. Inverse Functions
Do not confuse reciprocal functions (csc, sec, cot) with inverse trigonometric functions (arcsin, arccos, arctan). Cosecant is 1/sin(θ), whereas arcsin(x) is the angle whose sine is x. They are fundamentally different operations.

Correction Examples

  • Incorrect: cos(180) = -0.598 (calculated in radians). Correct: cos(180°) = -1 (calculated in degrees).
  • sec(90°) is undefined because it is 1/cos(90°).

Mathematical Derivations and Properties

  • Key Trigonometric Identities
  • Periodicity and Symmetry of Functions
  • Calculus of Trigonometric Functions
Deeper understanding comes from exploring the mathematical properties and identities that govern trigonometric functions.
The Pythagorean Identity
The cornerstone of trigonometric identities is sin²(θ) + cos²(θ) = 1. This identity is derived directly from the Pythagorean theorem (a² + b² = c²) applied to a right triangle within the unit circle. It leads to two other important identities: 1 + tan²(θ) = sec²(θ) and 1 + cot²(θ) = csc²(θ).
Periodicity and Even/Odd Properties
All trigonometric functions are periodic. Sine, cosine, cosecant, and secant have a period of 360° (2π radians), meaning their values repeat every 360°. Tangent and cotangent have a period of 180° (π radians). Cosine and secant are even functions (cos(-θ) = cos(θ)), while the other four are odd functions (e.g., sin(-θ) = -sin(θ)).
Derivatives in Calculus
In calculus, the derivatives of sine and cosine are fundamental: d/dx(sin(x)) = cos(x) and d/dx(cos(x)) = -sin(x). These relationships are crucial for studying rates of change in cyclical processes.

Property Examples

  • sin²(45°) + cos²(45°) = (√2/2)² + (√2/2)² = 0.5 + 0.5 = 1
  • The graph of y = cos(x) is symmetric about the y-axis.