Trigonometric Equation Solver
Working Backward from a Ratio to an Angle
Most trig calculators go from an angle to a ratio. This one runs the process in reverse: given that sin(x), cos(x), or tan(x) equals a specific value, it solves for x. Because sine, cosine, and tangent are periodic, more than one angle in a full circle can satisfy the same equation, so the calculator returns every solution within a single 360° rotation rather than just one.
The Formulas
sin(x) = value → x = arcsin(value) or x = 180° − arcsin(value)
cos(x) = value → x = arccos(value) or x = 360° − arccos(value)
tan(x) = value → x = arctan(value) or x = arctan(value) + 180°
cos(x) = value → x = arccos(value) or x = 360° − arccos(value)
tan(x) = value → x = arctan(value) or x = arctan(value) + 180°
Sine and cosine require an input value between −1 and 1, since neither ratio can exceed that range; tangent has no such restriction, as it can take any real value.
Where This Is Used
- Physics problem-solving — finding a launch angle or incidence angle from a known ratio of forces or velocities.
- Engineering design — solving for the angle that produces a required slope or force component.
- Trigonometry coursework — verifying answers to "solve for x" problems across all valid solutions in a given range.
| Equation | Solutions |
|---|---|
| sin(x) = 0.5 | x = 30°, 150° |
| cos(x) = 0.5 | x = 60°, 300° |
| tan(x) = 1 | x = 45°, 225° |
How to Use This Calculator
- Choose the function: sin(x) = value, cos(x) = value, or tan(x) = value.
- Enter the target Value.
- Select Calculate to see every solution for x between 0° and 360°.
Related Calculations
Need the forward calculation instead — angle to ratio? See the Sine Calculator, Cosine Calculator, or Tangent Calculator.