Trigonometry is a fundamental branch of mathematics that studies the relationships between the angles and sides of triangles. The six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) are central to this study and have widespread applications in various fields, including physics, engineering, navigation, astronomy, and computer graphics.
The Core Functions: Sine, Cosine, and Tangent
In a right-angled triangle, these functions are defined based on the ratios of its sides relative to one of its non-right angles (let's call it θ):
Sine (sin θ): The ratio of the length of the side opposite the angle to the length of the hypotenuse. (sin θ = Opposite / Hypotenuse)
Cosine (cos θ): The ratio of the length of the adjacent side to the angle to the length of the hypotenuse. (cos θ = Adjacent / Hypotenuse)
Tangent (tan θ): The ratio of the length of the side opposite the angle to the length of the adjacent side. (tan θ = Opposite / Adjacent)
These three are often considered the primary trigonometric functions, as the other three are their reciprocals.
The Reciprocal Functions: Cosecant, Secant, and Cotangent
The remaining three functions are defined as the reciprocals of the primary three:
Cosecant (csc θ): The reciprocal of sine. (csc θ = 1 / sin θ = Hypotenuse / Opposite)
Secant (sec θ): The reciprocal of cosine. (sec θ = 1 / cos θ = Hypotenuse / Adjacent)
Cotangent (cot θ): The reciprocal of tangent. (cot θ = 1 / tan θ = Adjacent / Opposite)
Working with Angles in Degrees and Radians
Trigonometric functions can operate on angles measured in degrees or radians. A full circle is 360 degrees, which is equivalent to 2π radians. This calculator allows you to specify which unit you are using for your input angle.
Conversion: To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
How the Calculator Works
This calculator takes an angle value and its unit (degrees or radians) as input. It then computes the values of all six trigonometric functions for that angle. For inputs that might lead to undefined results (like the tangent of 90 degrees or cosine of 0 degrees for cotangent/cosecant), the calculator will indicate this.
Engineering: Designing structures, analyzing forces, signal processing, and control systems.
Navigation: Calculating positions, distances, and bearings using celestial or terrestrial references.
Computer Graphics: Creating rotations, animations, and rendering 3D environments.
Surveying: Measuring distances and elevations indirectly.
Understanding and utilizing these functions is crucial for solving problems involving periodic phenomena, cyclical patterns, and geometric relationships.
function calculateTrigFunctions() {
var angleInput = document.getElementById("angleValue");
var unitSelect = document.getElementById("angleUnit");
var angleValue = parseFloat(angleInput.value);
var unit = unitSelect.value;
var resultSinSpan = document.getElementById("resultSin");
var resultCosSpan = document.getElementById("resultCos");
var resultTanSpan = document.getElementById("resultTan");
var resultCscSpan = document.getElementById("resultCsc");
var resultSecSpan = document.getElementById("resultSec");
var resultCotSpan = document.getElementById("resultCot");
// Clear previous results
resultSinSpan.textContent = "-";
resultCosSpan.textContent = "-";
resultTanSpan.textContent = "-";
resultCscSpan.textContent = "-";
resultSecSpan.textContent = "-";
resultCotSpan.textContent = "-";
if (isNaN(angleValue)) {
alert("Please enter a valid number for the angle value.");
return;
}
var angleInRadians;
if (unit === "degrees") {
angleInRadians = angleValue * (Math.PI / 180);
} else {
angleInRadians = angleValue;
}
// Calculations
var sinValue = Math.sin(angleInRadians);
var cosValue = Math.cos(angleInRadians);
var tanValue = Math.tan(angleInRadians);
// Handle division by zero for reciprocal functions
var cscValue = (sinValue === 0) ? "Undefined (sin is 0)" : (1 / sinValue);
var secValue = (cosValue === 0) ? "Undefined (cos is 0)" : (1 / cosValue);
var cotValue = (tanValue === 0) ? "Undefined (tan is 0)" : (1 / tanValue);
// Display results, formatting to a reasonable number of decimal places
resultSinSpan.textContent = typeof sinValue === 'number' ? sinValue.toFixed(6) : sinValue;
resultCosSpan.textContent = typeof cosValue === 'number' ? cosValue.toFixed(6) : cosValue;
resultTanSpan.textContent = typeof tanValue === 'number' ? tanValue.toFixed(6) : tanValue;
resultCscSpan.textContent = typeof cscValue === 'number' ? cscValue.toFixed(6) : cscValue;
resultSecSpan.textContent = typeof secValue === 'number' ? secValue.toFixed(6) : secValue;
resultCotSpan.textContent = typeof cotValue === 'number' ? cotValue.toFixed(6) : cotValue;
}