Trigonometric Equation Solver | sin, cos & tan Equations
Solve trigonometric equations in the forms a·sin(bx+c)+d=0, a·cos(bx+c)+d=0, a·tan(bx+c)+d=0, and quadratic-in-trig forms like a·sin²x+b·sinx+c=0. Returns exact values at standard angles, the general solution in terms of n, and all solutions in [0, 2π].
Trig function
Form: a·sin(bx+c) + d = 0
What Is the Trigonometric Equation Solver | sin, cos & tan Equations?
A trigonometric equation contains one or more trig functions (sin, cos, tan) of the unknown x. Linear trig equations isolate the trig function and apply the appropriate inverse. Quadratic-in-trig equations (like 2sin²x−sinx−1=0) use the quadratic formula to find the trig values, then solve each linear sub-equation. The period of sin and cos is 2π, so there are infinitely many solutions — expressed as a general formula involving integer n.
Formula
Linear: a·f(bx + c) + d = 0
Quadratic: A·f²(x) + B·f(x) + C = 0
sin: x = (arcsin(v) − c) / b + 2πn/b
cos: x = (±arccos(v) − c) / b + 2πn/b
tan: x = (arctan(v) − c) / b + πn/b
How to Use
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Select Linear or Quadratic equation type
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Choose the trig function: sin, cos, or tan
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Enter coefficients a, b, c, d (linear) or A, B, C (quadratic)
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Or load a preset to see a worked example
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Click Solve to compute the general solution
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Read exact values at standard angles and all solutions in [0, 2π]
Choose between a linear equation a·f(bx+c)+d=0 or a quadratic equation A·f²(x)+B·f(x)+C=0. Select the trig function. Enter the coefficients. Click Solve to get the general solution and all specific solutions in [0, 2π].
Example Calculation
Example: 2sin(x) − 1 = 0
Rearrange: sin(x) = 1/2
Arcsin(1/2) = π/6 (exact value)
General: x = π/6 + 2πn or x = 5π/6 + 2πn
In [0, 2π]: x = π/6 ≈ 0.524 rad and x = 5π/6 ≈ 2.618 rad
Frequently Asked Questions
Why are there infinitely many solutions?
Trig functions are periodic — sin(x) = sin(x+2π), cos(x) = cos(x+2π), tan(x) = tan(x+π). So if x₀ is a solution, then x₀ + 2πn (or x₀ + πn for tan) is also a solution for every integer n. The general solution captures all of them; choosing specific values of n gives solutions in a desired interval like [0, 2π].
How does the solver handle quadratic trig equations?
An equation like 2sin²x − sinx − 1 = 0 is quadratic in sin(x). Let u = sinx: 2u²−u−1=0, which factors or is solved with the quadratic formula to get u = 1 or u = −1/2. Each valid value (in [−1, 1] for sin/cos, any real for tan) becomes a linear trig equation solved separately.
What are exact values?
Many inverse trig values at common arguments are expressible as simple fractions of π: arcsin(0)=0, arcsin(1/2)=π/6, arcsin(√2/2)=π/4, arcsin(√3/2)=π/3, arcsin(1)=π/2. The solver recognizes these and returns the exact symbolic form rather than a decimal approximation.
What does no solution mean?
For sin and cos, the argument must lie in [−1, 1]. If the equation reduces to sin(x)=2, there is no real solution. For tan, any real value is valid. When the quadratic formula gives complex roots, those branches have no real solutions.
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