A trigonometric equation is an equation that involves one or more trigonometric functions of an unknown angle. The goal of this calculator is to find all possible values of x that satisfy the equation in the form f(x) = a.
Primary Formulas Used
Since trigonometric functions are periodic, they have an infinite number of solutions. We provide the "Principal Solutions" (usually within the first period) and the "General Solution" format.
For sin(x) = a: x = arcsin(a) + 2kπ and x = (π – arcsin(a)) + 2kπ
For cos(x) = a: x = arccos(a) + 2kπ and x = -arccos(a) + 2kπ
For tan(x) = a: x = arctan(a) + kπ
Example Calculation
Suppose you want to solve sin(x) = 0.5 in Degrees:
Select sin(x) from the dropdown.
Enter 0.5 as the value.
Select Degrees as the unit.
The calculator will provide 30° and 150° as the primary solutions within the 0° to 360° range.
Domain and Range Constraints
It is important to remember that for sin(x) and cos(x), the value of a must fall within the interval [-1, 1]. If you enter a value like 1.5, the equation will have no real solutions because the sine and cosine of an angle can never exceed 1 or be less than -1 on the unit circle. tan(x), however, can equal any real number.
function calculateTrig() {
var func = document.getElementById("trigFunc").value;
var a = parseFloat(document.getElementById("valA").value);
var unit = document.getElementById("unitType").value;
var resultDiv = document.getElementById("trigResult");
resultDiv.style.display = "block";
resultDiv.style.backgroundColor = "#f0f7fb";
resultDiv.style.border = "1px solid #3498db";
if (isNaN(a)) {
resultDiv.innerHTML = "Please enter a valid numeric value for 'a'.";
return;
}
var sol1, sol2, general;
var outputHtml = "
Solutions:
";
if (func === "sin") {
if (a 1) {
resultDiv.innerHTML = "No Real Solution: For sin(x), 'a' must be between -1 and 1.";
return;
}
var rad1 = Math.asin(a);
var rad2 = Math.PI – rad1;
if (unit === "degrees") {
sol1 = (rad1 * 180 / Math.PI).toFixed(2);
sol2 = (rad2 * 180 / Math.PI).toFixed(2);
outputHtml += "Primary Solutions (0° to 360°): " + sol1 + "° and " + sol2 + "°";
outputHtml += "General Solution: " + sol1 + "° + 360°k and " + sol2 + "° + 360°k";
} else {
outputHtml += "Primary Solutions (0 to 2π): " + rad1.toFixed(4) + " rad and " + rad2.toFixed(4) + " rad";
outputHtml += "General Solution: " + rad1.toFixed(4) + " + 2kπ and " + rad2.toFixed(4) + " + 2kπ";
}
}
else if (func === "cos") {
if (a 1) {
resultDiv.innerHTML = "No Real Solution: For cos(x), 'a' must be between -1 and 1.";
return;
}
var rad1 = Math.acos(a);
var rad2 = (2 * Math.PI) – rad1;
if (rad2 >= 2 * Math.PI) rad2 = rad1; // Handle edge case for 1 and -1
if (unit === "degrees") {
sol1 = (rad1 * 180 / Math.PI).toFixed(2);
sol2 = (360 – sol1).toFixed(2);
if (sol1 == 0 || sol1 == 180) {
outputHtml += "Primary Solution: " + sol1 + "°";
} else {
outputHtml += "Primary Solutions (0° to 360°): " + sol1 + "° and " + sol2 + "°";
}
outputHtml += "General Solution: ±" + sol1 + "° + 360°k";
} else {
outputHtml += "Primary Solutions (0 to 2π): " + rad1.toFixed(4) + " rad and " + rad2.toFixed(4) + " rad";
outputHtml += "General Solution: ±" + rad1.toFixed(4) + " + 2kπ";
}
}
else if (func === "tan") {
var rad1 = Math.atan(a);
var rad2 = rad1 + Math.PI;
if (unit === "degrees") {
sol1 = (rad1 * 180 / Math.PI).toFixed(2);
var sol1Pos = (parseFloat(sol1) < 0) ? (parseFloat(sol1) + 180).toFixed(2) : sol1;
var sol2Pos = (parseFloat(sol1Pos) + 180).toFixed(2);
outputHtml += "Primary Solutions (0° to 360°): " + sol1Pos + "° and " + sol2Pos + "°";
outputHtml += "General Solution: " + sol1 + "° + 180°k";
} else {
var rad1Pos = (rad1 < 0) ? (rad1 + Math.PI) : rad1;
var rad2Pos = rad1Pos + Math.PI;
outputHtml += "Primary Solutions (0 to 2π): " + rad1Pos.toFixed(4) + " rad and " + rad2Pos.toFixed(4) + " rad";
outputHtml += "General Solution: " + rad1.toFixed(4) + " + kπ";
}
}
outputHtml += "*where k is any integer (…-1, 0, 1…)";
resultDiv.innerHTML = outputHtml;
}