Cosine Rule Calculator

by

Cosine Rule Calculator

Use this calculator to find an unknown side length or an unknown angle of a triangle using the Cosine Rule.

Side c (given sides a, b and angle C) Angle C (given sides a, b, and c)

Calculate Side c

Enter two side lengths and the included angle.

Calculate Angle C

Enter all three side lengths.

Result:

function toggleInputFields() { var type = document.getElementById("calculationType").value; if (type === "side") { document.getElementById("calcSideInputs").style.display = "block"; document.getElementById("calcAngleInputs").style.display = "none"; } else { document.getElementById("calcSideInputs").style.display = "none"; document.getElementById("calcAngleInputs").style.display = "block"; } document.getElementById("cosineResult").innerHTML = ""; // Clear previous result } function calculateCosineRule() { var type = document.getElementById("calculationType").value; var resultDiv = document.getElementById("cosineResult"); resultDiv.innerHTML = ""; // Clear previous result if (type === "side") { var a = parseFloat(document.getElementById("sideA_calcSide").value); var b = parseFloat(document.getElementById("sideB_calcSide").value); var angleC_deg = parseFloat(document.getElementById("angleC_calcSide").value); if (isNaN(a) || isNaN(b) || isNaN(angleC_deg) || a <= 0 || b <= 0 || angleC_deg = 180) { resultDiv.innerHTML = "Please enter valid positive numbers for sides and an angle between 0 and 180 degrees."; return; } var angleC_rad = angleC_deg * Math.PI / 180; var c_squared = a * a + b * b – 2 * a * b * Math.cos(angleC_rad); if (c_squared < 0) { resultDiv.innerHTML = "Error: Cannot calculate side c with given inputs. This might indicate an invalid triangle configuration."; return; } var c = Math.sqrt(c_squared); resultDiv.innerHTML = "The length of Side c is approximately: " + c.toFixed(4) + ""; } else { // type === "angle" var a = parseFloat(document.getElementById("sideA_calcAngle").value); var b = parseFloat(document.getElementById("sideB_calcAngle").value); var c = parseFloat(document.getElementById("sideC_calcAngle").value); if (isNaN(a) || isNaN(b) || isNaN(c) || a <= 0 || b <= 0 || c c) && (a + c > b) && (b + c > a))) { resultDiv.innerHTML = "Error: The given side lengths do not form a valid triangle (Triangle Inequality Theorem)."; return; } var cosC = (a * a + b * b – c * c) / (2 * a * b); // Check if cosC is within valid range [-1, 1] due to floating point inaccuracies if (cosC > 1) cosC = 1; if (cosC < -1) cosC = -1; var angleC_rad = Math.acos(cosC); var angleC_deg = angleC_rad * 180 / Math.PI; resultDiv.innerHTML = "The measure of Angle C is approximately: " + angleC_deg.toFixed(4) + " degrees"; } } // Initialize the correct input fields on page load window.onload = function() { toggleInputFields(); }; .cosine-rule-calculator { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; font-family: Arial, sans-serif; } .cosine-rule-calculator h2, .cosine-rule-calculator h3 { color: #333; text-align: center; margin-top: 10px; } .calculator-inputs label { display: block; margin-bottom: 5px; font-weight: bold; color: #555; } .calculator-inputs input[type="number"], .calculator-inputs select { width: calc(100% – 22px); padding: 10px; margin-bottom: 15px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculator-inputs button { background-color: #007bff; color: white; padding: 12px 20px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; width: 100%; box-sizing: border-box; } .calculator-inputs button:hover { background-color: #0056b3; } .calculator-result { margin-top: 20px; padding: 15px; background-color: #e9ecef; border: 1px solid #dee2e6; border-radius: 4px; text-align: center; } .calculator-result p { margin: 0; font-size: 1.1em; color: #333; } .calculator-result strong { color: #007bff; } .calculator-inputs div { margin-top: 15px; padding-top: 10px; border-top: 1px dashed #eee; } .calculator-inputs div:first-of-type { border-top: none; padding-top: 0; }

Understanding the Cosine Rule

The Cosine Rule, also known as the Law of Cosines, is a fundamental theorem in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. It is a generalization of the Pythagorean theorem, which only applies to right-angled triangles. The Cosine Rule is particularly useful when you need to find an unknown side or an unknown angle in a non-right-angled triangle.

When to Use the Cosine Rule

The Cosine Rule is typically applied in two main scenarios:

  1. Side-Angle-Side (SAS): When you know two sides of a triangle and the included angle (the angle between those two sides), you can use the Cosine Rule to find the length of the third side.
  2. Side-Side-Side (SSS): When you know the lengths of all three sides of a triangle, you can use the Cosine Rule to find the measure of any of the angles.

The Formulas

For a triangle with sides a, b, c and angles A, B, C opposite to those sides respectively, the Cosine Rule can be stated as:

  • To find a side:
    • c² = a² + b² - 2ab cos(C)
    • a² = b² + c² - 2bc cos(A)
    • b² = a² + c² - 2ac cos(B)
  • To find an angle:
    • cos(C) = (a² + b² - c²) / (2ab)
    • cos(A) = (b² + c² - a²) / (2bc)
    • cos(B) = (a² + c² - b²) / (2ac)

Our calculator focuses on finding side 'c' or angle 'C' using the first formula in each set, but the principle applies to any side or angle by relabeling the triangle.

How to Use the Calculator

This Cosine Rule Calculator simplifies the process of applying these formulas. Follow these steps:

  1. Select Calculation Type: Choose whether you want to "Calculate Side c" or "Calculate Angle C" from the dropdown menu.
  2. Enter Values:
    • For Side c: Input the lengths of Side a, Side b, and the measure of Angle C (in degrees).
    • For Angle C: Input the lengths of Side a, Side b, and Side c.
  3. Click "Calculate": The calculator will instantly display the unknown side length or angle measure.

Examples

Example 1: Calculating a Side (SAS)

Imagine you have a triangle where:

  • Side a = 10 units
  • Side b = 12 units
  • Included Angle C = 60 degrees

You want to find the length of Side c.

Using the formula c² = a² + b² - 2 * a * b * cos(C):

c² = 10² + 12² - 2 * 10 * 12 * cos(60°)

c² = 100 + 144 - 240 * 0.5

c² = 244 - 120

c² = 124

c = √124 ≈ 11.1355 units

Using the calculator with these values will yield approximately 11.1355.

Example 2: Calculating an Angle (SSS)

Consider a triangle with the following side lengths:

  • Side a = 7 units
  • Side b = 8 units
  • Side c = 9 units

You want to find the measure of Angle C.

Using the formula cos(C) = (a² + b² - c²) / (2ab):

cos(C) = (7² + 8² - 9²) / (2 * 7 * 8)

cos(C) = (49 + 64 - 81) / (112)

cos(C) = (113 - 81) / 112

cos(C) = 32 / 112

cos(C) = 0.285714...

C = arccos(0.285714...) ≈ 73.4007 degrees

Using the calculator with these values will yield approximately 73.4007 degrees.

Importance and Applications

The Cosine Rule is a powerful tool with wide-ranging applications in various fields:

  • Geometry and Surveying: Used to calculate distances and angles in land surveying, mapping, and construction, especially when direct measurement is difficult or impossible.
  • Navigation: Essential for calculating distances and bearings in air and sea navigation.
  • Engineering: Applied in structural analysis, robotics, and mechanical design to determine forces, stresses, and positions.
  • Physics: Used in vector addition and resolving forces, particularly in problems involving non-perpendicular vectors.
  • Computer Graphics: Fundamental for calculations in 3D modeling and animation, such as determining angles between surfaces or positions of objects.

By providing a method to solve triangles when the Pythagorean theorem isn't applicable, the Cosine Rule significantly expands our ability to analyze and understand geometric relationships in the real world.

Leave a Comment