6 Trigonometric Functions Calculator
Calculate sine, cosine, tangent, cosecant, secant, and cotangent for any angle with our 6 Trigonometric Functions Calculator.
Trigonometric Calculator
Results
csc(θ) = 1/sin(θ), sec(θ) = 1/cos(θ), cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)
Results Summary Table
| Function | Value |
|---|---|
| Sine (sin θ) | – |
| Cosine (cos θ) | – |
| Tangent (tan θ) | – |
| Cosecant (csc θ) | – |
| Secant (sec θ) | – |
| Cotangent (cot θ) | – |
Summary of the 6 trigonometric function values for the given angle.
Function Values Visualization
Bar chart showing the absolute values of Sine, Cosine, and Tangent (clipped at 5 for very large tan values). Values greater than 5 are shown as 5.
What is a 6 Trigonometric Functions Calculator?
A 6 Trigonometric Functions Calculator is a tool designed to compute the values of the six fundamental trigonometric functions for a given angle. These functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). The calculator accepts an angle input, either in degrees or radians, and outputs the corresponding values for all six functions.
This calculator is useful for students learning trigonometry, engineers, scientists, and anyone working with angles and their relationships in triangles or periodic phenomena. It helps in quickly finding the ratios of sides in a right-angled triangle or coordinates on a unit circle associated with a specific angle.
Common misconceptions include thinking these functions only apply to right-angled triangles; in reality, they are defined for any angle using the unit circle and are fundamental to understanding waves, oscillations, and many other areas of mathematics and physics.
6 Trigonometric Functions Formula and Mathematical Explanation
The six trigonometric functions relate the angles of a right triangle to the ratios of the lengths of its sides, or more generally, relate an angle to the coordinates of a point on a unit circle centered at the origin.
For an angle θ, consider a point (x, y) on a unit circle (radius r=1) corresponding to θ, or a right-angled triangle with angle θ, opposite side 'o', adjacent side 'a', and hypotenuse 'h'.
- Sine (sin θ): y/r (on unit circle) or o/h (in triangle)
- Cosine (cos θ): x/r (on unit circle) or a/h (in triangle)
- Tangent (tan θ): y/x (on unit circle) or o/a (in triangle) = sin θ / cos θ
- Cosecant (csc θ): r/y (on unit circle) or h/o (in triangle) = 1 / sin θ
- Secant (sec θ): r/x (on unit circle) or h/a (in triangle) = 1 / cos θ
- Cotangent (cot θ): x/y (on unit circle) or a/o (in triangle) = 1 / tan θ = cos θ / sin θ
When using the unit circle (r=1), sin θ = y, cos θ = x, tan θ = y/x, csc θ = 1/y, sec θ = 1/x, and cot θ = x/y.
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| θ | Angle | Degrees or Radians | Any real number |
| sin θ, cos θ | Sine and Cosine values | Dimensionless ratio | -1 to 1 |
| tan θ, cot θ | Tangent and Cotangent values | Dimensionless ratio | -∞ to ∞ (undefined at certain points) |
| csc θ, sec θ | Cosecant and Secant values | Dimensionless ratio | (-∞, -1] U [1, ∞) (undefined at certain points) |
Practical Examples (Real-World Use Cases)
Example 1: Right-Angled Triangle
Suppose you have a right-angled triangle where one angle is 30 degrees and the hypotenuse is 10 units. You want to find the lengths of the opposite and adjacent sides.
- sin(30°) = opposite/10 => opposite = 10 * sin(30°) = 10 * 0.5 = 5 units
- cos(30°) = adjacent/10 => adjacent = 10 * cos(30°) = 10 * (√3/2) ≈ 8.66 units
Our 6 Trigonometric Functions Calculator quickly gives sin(30°) = 0.5 and cos(30°) ≈ 0.866.
Example 2: Wave Analysis
A simple wave can be described by y = A sin(ωt + φ). If you know the phase angle (ωt + φ) at a certain time, say 60 degrees, you can find the displacement 'y' relative to the amplitude 'A'.
- y = A sin(60°) = A * (√3/2) ≈ 0.866 A
Using the 6 Trigonometric Functions Calculator for 60 degrees gives sin(60°) ≈ 0.866.
How to Use This 6 Trigonometric Functions Calculator
- Enter the Angle Value: Input the numerical value of the angle into the "Angle Value" field.
- Select the Angle Unit: Choose whether the entered angle is in "Degrees" or "Radians" using the radio buttons.
- View Results: The calculator automatically updates and displays the values of sine, cosine, tangent, cosecant, secant, and cotangent in the "Results" section as you type or change the unit.
- Check the Table and Chart: The "Results Summary Table" and the "Function Values Visualization" chart also update dynamically.
- Reset: Click the "Reset" button to clear the input and results to their default values (30 degrees).
- Copy Results: Click "Copy Results" to copy the angle, unit, and the six function values to your clipboard.
The results show the direct values. "Undefined" or "Infinity" will be displayed for functions like tan(90°), csc(0°), etc., where the denominator is zero.
Key Factors That Affect 6 Trigonometric Functions Calculator Results
- Angle Value: The primary input that determines the output values. Different angles yield different function values.
- Angle Unit: Whether the angle is measured in degrees or radians significantly changes the input for the mathematical functions (e.g., sin(30 degrees) is different from sin(30 radians)).
- Quadrant of the Angle: The signs (+ or -) of the trigonometric functions depend on which quadrant (I, II, III, or IV) the angle lies in.
- Proximity to Asymptotes: For tan, cot, sec, and csc, angles close to where the functions are undefined (e.g., tan near 90°, csc near 0°) will result in very large positive or negative values.
- Precision of π: When converting between degrees and radians, the precision of the value used for π can slightly affect results for radian calculations or degree-to-radian conversions. Our calculator uses `Math.PI`.
- Floating-Point Precision: Computers use floating-point arithmetic, which can introduce very small rounding errors in the results, especially for angles that should yield exact zeros or ones for certain functions after many operations.
Frequently Asked Questions (FAQ)
- What are the 6 trigonometric functions?
- Sine (sin), Cosine (cos), Tangent (tan), Cosecant (csc), Secant (sec), and Cotangent (cot).
- What is the difference between degrees and radians?
- Both are units for measuring angles. A full circle is 360 degrees or 2π radians. 180 degrees = π radians.
- Why is tan(90°) undefined?
- Because tan(θ) = sin(θ)/cos(θ), and cos(90°) = 0. Division by zero is undefined.
- What is the range of sine and cosine?
- The values of sine and cosine range from -1 to +1, inclusive.
- What is the range of secant and cosecant?
- The values of secant and cosecant are always less than or equal to -1, or greater than or equal to +1. They are never between -1 and 1 (exclusive).
- How do I use the 6 Trigonometric Functions Calculator for a negative angle?
- Simply enter the negative value in the "Angle Value" field. The calculator will compute the functions based on the negative angle (e.g., sin(-30°) = -sin(30°)).
- Can I find the angle from a trigonometric value using this calculator?
- No, this 6 Trigonometric Functions Calculator finds the function values from the angle. For the reverse, you would need an inverse trigonometric function calculator (arcsin, arccos, arctan).
- Where are trigonometric functions used?
- They are used in physics (waves, optics, mechanics), engineering (structural analysis, electronics), navigation, astronomy, computer graphics, and many other fields.