Accurately compute Sine, Tangent, and Cosine for any angle with our intuitive tool.
Trigonometric Function Calculator
Enter the angle in Degrees (°)
Degrees (°)
Radians
Select the unit for your angle input.
Results
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Calculations based on standard trigonometric identities.
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Sine
—
Cosine
—
Tangent
Results copied!
Trigonometric Values Table
Angle (Degrees)
Angle (Radians)
Sine
Cosine
Tangent
A sample of trigonometric values for common angles.
Trigonometric Function Graph
Visual representation of Sine and Cosine functions.
What is a Sin Tan Cos Calculator?
A Sin Tan Cos calculator is a specialized tool designed to compute the sine (sin), tangent (tan), and cosine (cos) values for a given angle. These functions are fundamental components of trigonometry, a branch of mathematics concerned with the relationships between the sides and angles of triangles, and with trigonometric functions and their periodic behavior. This calculator allows users to input an angle in degrees or radians and instantly obtain the corresponding sine, cosine, and tangent values. It's an indispensable tool for students, educators, engineers, physicists, and anyone working with geometric calculations or wave phenomena.
Who Should Use It?
This calculator is beneficial for a wide range of users:
Students: High school and college students learning trigonometry, geometry, calculus, and physics.
Educators: Teachers who need to demonstrate trigonometric concepts or provide quick calculations for their students.
Engineers: Particularly those in civil, mechanical, electrical, and aerospace engineering, where trigonometry is crucial for structural analysis, circuit design, and motion studies.
Physicists: For calculations involving waves, oscillations, optics, and mechanics.
Surveyors and Navigators: For calculating distances, heights, and positions.
Computer Graphics Professionals: For 2D and 3D transformations, rotations, and animations.
Common Misconceptions
One common misconception is that sine, tangent, and cosine are only relevant to right-angled triangles. While they are first introduced in that context, these functions extend to all angles, positive and negative, and are crucial for understanding periodic functions and wave mechanics in broader mathematical and scientific fields. Another is confusion between degrees and radians; this calculator helps clarify that by allowing unit selection.
Sin Tan Cos Calculator Formula and Mathematical Explanation
The sine, cosine, and tangent functions are defined using the unit circle and right-angled triangles. On the unit circle (a circle with radius 1 centered at the origin), for an angle θ measured counterclockwise from the positive x-axis, the coordinates of the point where the terminal side of the angle intersects the circle are (cos θ, sin θ). The tangent is then defined as the ratio of sine to cosine.
Unit Circle Definition
For an angle θ:
Sine (sin θ): The y-coordinate of the point on the unit circle.
Cosine (cos θ): The x-coordinate of the point on the unit circle.
Tangent (tan θ): The ratio of the sine to the cosine, tan θ = sin θ / cos θ. This represents the slope of the line segment from the origin to the point on the unit circle.
Right-Angled Triangle Definition
In a right-angled triangle, for an acute angle θ:
Sine (sin θ): Opposite side / Hypotenuse
Cosine (cos θ): Adjacent side / Hypotenuse
Tangent (tan θ): Opposite side / Adjacent side
The calculator uses these fundamental definitions, applying them to the input angle value, converting between degrees and radians as needed using the conversion factor: π radians = 180 degrees.
Variables Table
Variable
Meaning
Unit
Typical Range
θ (Theta)
The angle for which trigonometric values are calculated.
Degrees or Radians
(-∞, ∞)
sin θ
The sine of the angle θ.
Unitless
[-1, 1]
cos θ
The cosine of the angle θ.
Unitless
[-1, 1]
tan θ
The tangent of the angle θ.
Unitless
(-∞, ∞)
Practical Examples (Real-World Use Cases)
Example 1: Calculating Height Using an Angle of Elevation
Imagine you are standing 50 meters away from a tall building. You measure the angle of elevation from your eye level to the top of the building to be 30 degrees. You want to estimate the building's height. Assuming your eye level is 1.6 meters from the ground.
Input Angle: 30 Degrees
Distance (Adjacent side): 50 meters
Assumption: Eye level height = 1.6 meters
We use the tangent function, as tan(θ) = Opposite / Adjacent. The 'Opposite' side here is the height of the building above your eye level.
Total Building Height = Height_above_eye_level + Eye level height
Total Building Height ≈ 28.87m + 1.6m = 30.47 meters
Calculator Result: For an angle of 30 degrees, tan(30°) ≈ 0.5774. This value is used to find the height of the building.
Example 2: Analyzing Waveform Amplitude
In physics, sinusoidal waves (like sound waves or electromagnetic waves) are often described using sine functions. Suppose you are analyzing a simple harmonic motion where the position x(t) is given by x(t) = A * sin(ωt + φ). You need to find the position at a specific time. Let the amplitude A = 10 cm, angular frequency ω = π/2 rad/s, and phase angle φ = π/4 rad. You want to find the position at time t = 0.5 s.
Using the calculator (or knowing the unit circle), sin(π/2 radians) = 1.
Position x(0.5s) = 10 cm * sin(π/2 radians) = 10 cm * 1 = 10 cm.
Calculator Result: For an angle of π/2 radians (which is 90 degrees), sin(π/2) = 1. This indicates the object is at its maximum positive displacement.
How to Use This Sin Tan Cos Calculator
Using this trigonometric calculator is straightforward. Follow these steps:
Enter the Angle Value: Input the numerical value of the angle you wish to calculate the trigonometric functions for into the "Angle Value" field.
Select the Angle Unit: Choose whether your angle is measured in "Degrees (°)" or "Radians" using the dropdown menu.
Click 'Calculate': Press the "Calculate" button. The calculator will process your input and display the results.
How to Read Results
The calculator provides:
Primary Result (Sine): The largest, highlighted value shows the computed sine of your angle.
Intermediate Values (Cosine, Tangent): These are displayed below the main result, showing the computed cosine and tangent for your angle.
Table of Values: A table is generated showing the sine, cosine, and tangent for several common angles, including your input if it falls within the displayed range.
Graph: A visual representation plots the sine and cosine curves, helping you understand their behavior.
Decision-Making Guidance
The results from this calculator are foundational for many mathematical and scientific applications. Use the calculated values to:
Solve for unknown sides or angles in triangles.
Analyze wave patterns and periodic phenomena.
Perform geometric transformations in design and engineering.
Verify complex trigonometric identities.
The "Copy Results" button allows you to easily transfer the main and intermediate values for use in other documents or applications.
Key Factors That Affect Sin Tan Cos Results
While the sine, cosine, and tangent of an angle are primarily determined by the angle itself, several underlying factors and contexts influence their application and interpretation:
Angle Unit (Degrees vs. Radians): This is the most direct factor. The numerical value of a trigonometric function changes drastically depending on whether the angle is measured in degrees or radians. Using the wrong unit will yield incorrect results. For instance, sin(30°) = 0.5, while sin(30 radians) ≈ -0.988. Our sin tan cos calculator handles this conversion.
Quadrant of the Angle: The sign (+ or -) of sine, cosine, and tangent depends on the quadrant in which the angle's terminal side lies. Sine is positive in Quadrants I and II, negative in III and IV. Cosine is positive in I and IV, negative in II and III. Tangent is positive in I and III, negative in II and IV.
Periodicity of Functions: Sine and cosine are periodic functions with a period of 360° or 2π radians. This means sin(θ) = sin(θ + 360°n) and cos(θ) = cos(θ + 360°n) for any integer 'n'. Tangent has a period of 180° or π radians. This property is crucial in analyzing cyclical events.
Domain Restrictions (Tangent): The tangent function is undefined when cos(θ) = 0, which occurs at angles like 90°, 270°, and their equivalents (π/2, 3π/2 radians, etc.). At these points, the tangent approaches positive or negative infinity.
Context of Application: The relevance of a trigonometric value depends on the field. In engineering, a sine wave might represent AC voltage, while in physics, it could model simple harmonic motion. The interpretation changes, but the mathematical calculation remains the same. Understanding related concepts like frequency and amplitude is key.
Precision and Rounding: Calculators and software use finite precision. While this calculator aims for accuracy, results for certain irrational numbers (like tan(60°)) might be approximations. For highly sensitive applications, understanding numerical precision is important.
Frequently Asked Questions (FAQ)
What is the difference between degrees and radians?
Degrees and radians are two different units for measuring angles. A full circle is 360 degrees (360°). A full circle is also 2π radians. Radians are often preferred in higher mathematics and physics because they simplify many formulas, especially those involving calculus and trigonometric functions.
When is the tangent function undefined?
The tangent function (tan θ) is undefined when the cosine of the angle (cos θ) is zero. This occurs at angles that are odd multiples of 90 degrees (or π/2 radians), such as 90°, 270°, 450°, etc., and -90°, -270°, etc.
Why are sine and cosine values always between -1 and 1?
This is because sine and cosine are defined based on the coordinates of points on the unit circle (a circle with a radius of 1). The x and y coordinates of any point on a circle with radius 1 can never be further than 1 unit away from the center (origin). Therefore, cos θ (the x-coordinate) and sin θ (the y-coordinate) are always between -1 and 1, inclusive.
Can I calculate sin, cos, or tan for negative angles?
Yes, you can. Negative angles are measured clockwise from the positive x-axis. The trigonometric functions still apply. For example, sin(-45°) = -sin(45°), and cos(-45°) = cos(45°). This calculator accepts negative inputs.
How does this calculator handle angles greater than 360 degrees?
Sine and cosine functions are periodic with a period of 360 degrees (or 2π radians). This calculator, like standard trigonometric functions, will compute the value equivalent to the angle within the 0 to 360 degree range (or 0 to 2π radian range). For example, sin(405°) is the same as sin(45°).
What is the practical use of the tangent function?
The tangent function is particularly useful in calculating slopes, determining angles of elevation or depression, and solving problems involving right-angled triangles where the height and base are known. It represents the ratio of the opposite side to the adjacent side.
Can this calculator handle very large or very small angles?
Yes, within the limits of standard floating-point arithmetic. The mathematical definitions of sine, cosine, and tangent extend to all real numbers. The calculator will provide the corresponding values, though for extremely large angles, the periodicity becomes the dominant factor in the result's relation to smaller angles.
How accurate are the results?
The results are calculated using standard JavaScript `Math.sin()`, `Math.cos()`, and `Math.tan()` functions, which typically use double-precision floating-point numbers. This provides a high degree of accuracy suitable for most educational and practical applications. For highly specialized scientific or engineering tasks requiring arbitrary precision, dedicated libraries might be necessary.
A comprehensive resource for geometric formulas and principles, including trigonometry basics.
var angleInput = document.getElementById('angleValue');
var unitSelect = document.getElementById('angleUnit');
var sineOutput = document.getElementById('sineValue');
var cosineOutput = document.getElementById('cosineValue');
var tangentOutput = document.getElementById('tangentValue');
var mainResultOutput = document.getElementById('mainResult');
var trigTableBody = document.getElementById('trigTableBody');
var trigChartCanvas = document.getElementById('trigChart');
var trigChartInstance = null; // To hold the chart instance
// Function to convert degrees to radians
function degreesToRadians(degrees) {
return degrees * (Math.PI / 180);
}
// Function to format numbers for display
function formatNumber(num) {
if (isNaN(num)) return '–';
if (num === Infinity) return '∞';
if (num === -Infinity) return '-∞';
// Check for values very close to zero
if (Math.abs(num) < 1e-10) return '0';
return num.toFixed(6); // Adjust precision as needed
}
// Function to validate input
function validateInput() {
var angleError = document.getElementById('angleValueError');
var angleValue = parseFloat(angleInput.value);
angleError.textContent = ''; // Clear previous error
if (isNaN(angleValue)) {
angleError.textContent = 'Please enter a valid number.';
return false;
}
// No explicit range validation for angles as they can be any real number
return true;
}
// Function to update table
function updateTable() {
var rowsHtml = '';
var anglesToCalculate = [0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, 330, 360];
var currentAngleInput = parseFloat(angleInput.value);
var currentUnit = unitSelect.value;
// Add the user's input angle to the list if it's not already there, ensuring it's in degrees for sorting
var userAngleInDegrees = (currentUnit === 'degrees') ? currentAngleInput : currentAngleInput * (180 / Math.PI);
var isUserAngleIncluded = anglesToCalculate.some(function(deg) {
return Math.abs(deg – userAngleInDegrees) < 1e-6;
});
if (!isUserAngleIncluded) {
// Insert user angle while maintaining order, convert to degrees first
var tempAngles = anglesToCalculate.map(function(deg) { return { deg: deg, rad: degreesToRadians(deg) }; });
tempAngles.push({ deg: userAngleInDegrees, rad: degreesToRadians(userAngleInDegrees) });
tempAngles.sort(function(a, b) { return a.deg – b.deg; });
// Filter out duplicates and angles outside a reasonable viewing range if needed
var uniqueSortedAngles = [];
var seenDegrees = new Set();
tempAngles.forEach(function(item) {
if (!seenDegrees.has(item.deg)) {
uniqueSortedAngles.push(item);
seenDegrees.add(item.deg);
}
});
anglesToCalculate = uniqueSortedAngles.map(function(item) { return item.deg; });
}
for (var i = 0; i < anglesToCalculate.length; i++) {
var angleDeg = anglesToCalculate[i];
var angleRad = degreesToRadians(angleDeg);
var sinVal = Math.sin(currentUnit === 'degrees' ? degreesToRadians(angleDeg) : angleDeg);
var cosVal = Math.cos(currentUnit === 'degrees' ? degreesToRadians(angleDeg) : angleDeg);
var tanVal = Math.tan(currentUnit === 'degrees' ? degreesToRadians(angleDeg) : angleDeg);
// Special handling for 90 and 270 degrees for tangent to show 'undefined' or infinity
var tanDisplay = formatNumber(tanVal);
if (Math.abs(cosVal) < 1e-10) { // If cosine is close to zero
tanDisplay = "Undefined";
}
rowsHtml += '
';
rowsHtml += '
' + angleDeg.toFixed(1) + '
';
rowsHtml += '
' + formatNumber(angleRad) + '
';
rowsHtml += '
' + formatNumber(sinVal) + '
';
rowsHtml += '
' + formatNumber(cosVal) + '
';
rowsHtml += '
' + tanDisplay + '
';
rowsHtml += '
';
}
trigTableBody.innerHTML = rowsHtml;
}
// Function to update chart
function updateChart() {
if (trigChartInstance) {
trigChartInstance.destroy(); // Destroy previous chart if it exists
}
var ctx = trigChartCanvas.getContext('2d');
var angleDegrees = parseFloat(angleInput.value);
var angleUnit = unitSelect.value;
var chartLabels = [];
var sineData = [];
var cosineData = [];
var startAngleRad, endAngleRad;
var step = Math.PI / 36; // Step for chart rendering (e.g., 5 degrees step)
if (angleUnit === 'degrees') {
startAngleRad = degreesToRadians(angleDegrees – 180); // Show a range around the input
endAngleRad = degreesToRadians(angleDegrees + 180);
} else {
startAngleRad = angleDegrees – Math.PI;
endAngleRad = angleDegrees + Math.PI;
}
// Ensure a reasonable range if input is extreme
if (isNaN(startAngleRad) || isNaN(endAngleRad)) {
startAngleRad = degreesToRadians(-180);
endAngleRad = degreesToRadians(180);
}
for (var angleRad = startAngleRad; angleRad <= endAngleRad; angleRad += step) {
chartLabels.push(angleRad.toFixed(2) + " rad");
sineData.push(Math.sin(angleRad));
cosineData.push(Math.cos(angleRad));
}
trigChartInstance = new Chart(ctx, {
type: 'line',
data: {
labels: chartLabels,
datasets: [{
label: 'Sine',
data: sineData,
borderColor: 'rgba(75, 192, 192, 1)',
backgroundColor: 'rgba(75, 192, 192, 0.2)',
fill: false,
tension: 0.1
}, {
label: 'Cosine',
data: cosineData,
borderColor: 'rgba(255, 99, 132, 1)',
backgroundColor: 'rgba(255, 99, 132, 0.2)',
fill: false,
tension: 0.1
}]
},
options: {
responsive: true,
maintainAspectRatio: true, // Allow chart to adjust height
scales: {
x: {
title: {
display: true,
text: 'Angle (Radians)'
}
},
y: {
title: {
display: true,
text: 'Value'
},
min: -1.1,
max: 1.1
}
},
plugins: {
legend: {
position: 'top',
},
title: {
display: true,
text: 'Sine and Cosine Waveforms'
}
}
}
});
}
// Main calculation function
function calculateTrig() {
if (!validateInput()) {
return;
}
var angleValue = parseFloat(angleInput.value);
var unit = unitSelect.value;
var angleRad;
if (unit === 'degrees') {
angleRad = degreesToRadians(angleValue);
} else {
angleRad = angleValue;
}
var sinVal = Math.sin(angleRad);
var cosVal = Math.cos(angleRad);
var tanVal = Math.tan(angleRad);
// Check for undefined tangent
var tanDisplay = formatNumber(tanVal);
if (Math.abs(cosVal) < 1e-10) { // If cosine is close to zero
tanDisplay = "Undefined";
}
sineOutput.textContent = formatNumber(sinVal);
cosineOutput.textContent = formatNumber(cosVal);
tangentOutput.textContent = tanDisplay;
// Set main result to Sine by default
mainResultOutput.textContent = formatNumber(sinVal);
updateTable();
updateChart();
}
// Function to copy results
function copyResults() {
var sine = sineOutput.textContent;
var cosine = cosineOutput.textContent;
var tangent = tangentOutput.textContent;
var mainResult = mainResultOutput.textContent;
var angle = angleInput.value;
var unit = unitSelect.options[unitSelect.selectedIndex].text;
var resultText = "Trigonometric Calculation Results:\n\n";
resultText += "Input Angle: " + angle + " " + unit + "\n";
resultText += "————————————\n";
resultText += "Main Result (Sine): " + mainResult + "\n";
resultText += "Sine: " + sine + "\n";
resultText += "Cosine: " + cosine + "\n";
resultText += "Tangent: " + tangent + "\n";
// Use navigator.clipboard for modern browsers, fallback for older ones
if (navigator.clipboard && navigator.clipboard.writeText) {
navigator.clipboard.writeText(resultText).then(function() {
var copyMessage = document.getElementById('copyMessage');
copyMessage.style.display = 'block';
setTimeout(function() {
copyMessage.style.display = 'none';
}, 3000);
}).catch(function(err) {
console.error('Failed to copy text: ', err);
// Fallback or error message for the user
});
} else {
// Fallback for older browsers (less secure/reliable)
var textArea = document.createElement("textarea");
textArea.value = resultText;
textArea.style.position = "fixed"; // Avoid scrolling to bottom
textArea.style.left = "-9999px";
textArea.style.top = "-9999px";
document.body.appendChild(textArea);
textArea.focus();
textArea.select();
try {
var successful = document.execCommand('copy');
var msg = successful ? 'successful' : 'unsuccessful';
console.log('Fallback: Copying text command was ' + msg);
var copyMessage = document.getElementById('copyMessage');
copyMessage.style.display = 'block';
setTimeout(function() {
copyMessage.style.display = 'none';
}, 3000);
} catch (err) {
console.error('Fallback: Oops, unable to copy', err);
}
document.body.removeChild(textArea);
}
}
// Function to reset calculator
function resetCalculator() {
angleInput.value = '45';
unitSelect.value = 'degrees';
calculateTrig(); // Recalculate with default values
}
// Function to toggle FAQ content
function toggleFaq(element) {
var faqItem = element.closest('.faq-item');
faqItem.classList.toggle('active');
}
// Initial calculation on page load
document.addEventListener('DOMContentLoaded', function() {
// Add event listeners for real-time updates
angleInput.addEventListener('input', calculateTrig);
unitSelect.addEventListener('change', calculateTrig);
calculateTrig(); // Perform initial calculation
});
// Chart.js integration (assuming Chart.js library is available globally or included)
// If not, you would need to include it via CDN in the or before the script.
// For this example, we assume Chart.js is available.
// Example CDN:
// Ensure Chart.js is loaded before attempting to use it
if (typeof Chart === 'undefined') {
console.error("Chart.js library not found. Please include it in your HTML.");
// Optionally, you could dynamically load it or display a message to the user.
}