Solve A × f(Bx + C) = D
Enter the equation parameters. The calculator returns all real values inside your selected interval.
Example Data Table
This example solves 2 × sin(2x + 30) = 1 from 0° to 360°.
| Item | Value | Meaning |
|---|---|---|
| Function | sin | Use the sine inverse family. |
| A | 2 | Divide D by 2 before solving. |
| B | 2 | The internal angle changes twice as fast. |
| C | 30° | The internal angle includes a 30° phase value. |
| D | 1 | The reduced target becomes 0.5. |
| Interval | 0° to 360° | Only answers inside one requested cycle are shown. |
Formula Used
Start with A × f(Bx + C) = D. Divide by A to obtain f(Bx + C) = D ÷ A. Let q = D ÷ A and let θ = Bx + C.
- Sine: θ = asin(q) + 2πk or θ = π − asin(q) + 2πk.
- Cosine: θ = ±acos(q) + 2πk.
- Tangent: θ = atan(q) + πk.
- Cotangent: θ = atan(1 ÷ q) + πk, when q is not zero.
- Secant: solve cos(θ) = 1 ÷ q, then use the cosine family.
- Cosecant: solve sin(θ) = 1 ÷ q, then use the sine family.
- Final step: x = (θ − C) ÷ B, where k is any integer.
How to Use This Calculator
- Select sine, cosine, tangent, cotangent, secant, or cosecant.
- Select degrees or radians before entering phase and interval values.
- Enter A, B, C, and D from the equation A × f(Bx + C) = D.
- Set the start and end values for the domain you need.
- Choose displayed decimal places, then select Find solutions.
- Read the general solution first, then use the table for interval answers.
- Download the displayed table as CSV or print it as a PDF record.
Solving Trigonometric Equations
Understand the Equation
Trigonometric equations ask for angles or variables that make a trigonometric statement true. They appear in geometry, waves, engineering, and navigation. A good method prevents missed answers. Start by deciding whether the equation is already isolated. The calculator solves equations written as A × f(Bx + C) = D. Here, f is one trigonometric function. The variables A, B, C, and D are numeric values.
First divide both sides by A. This creates a simpler target value, q = D ÷ A. For sine and cosine, q must stay between negative one and one. Tangent and cotangent accept any real target. Secant and cosecant require a target whose magnitude is at least one. These checks are important. They show when a real solution cannot exist before any angle calculation begins.
Find Every Angle Family
Inverse functions give a principal angle. For example, arcsine returns one angle whose sine equals q. But trigonometric functions repeat. One principal angle is rarely the complete answer. Sine has two possible angle families in a full cycle. Cosine also has two families unless it reaches an endpoint. Tangent and cotangent repeat every half turn. The calculator creates every matching value inside the interval you enter.
A phase shift changes where a cycle begins. The expression Bx + C contains that shift. The frequency value B changes how quickly the cycle repeats. After finding each internal angle theta, solve the linear expression. Use x = (theta − C) ÷ B. This final algebra step is easy to overlook. It explains why listed x values may differ from familiar unit-circle angles.
Choose Units and Intervals
Choose degrees when your question uses degrees. Choose radians for calculus, scientific work, or formulas involving pi. Keep the interval in the same unit you selected. For example, enter 0 to 360 for one degree cycle. Enter 0 to 2π for one radian cycle. A narrow interval can contain none, even when the general solution exists.
The result panel shows the reduced target, the solution families, and all values found in your range. Review the equation displayed there first. Each row gives an x value and its related internal angle. Use the displayed residual check when you need extra confidence. A residual near zero means substituting the value returns both sides to nearly the same result.
Prepare and Check Your Work
Some equations need preparation before calculation. Use identities to rewrite products, squares, or mixed functions. Factor when possible. Move all terms to one side when that exposes a common factor. Then isolate one trigonometric expression. You can solve each factor separately. This calculator is most useful after the equation becomes one supported expression equal to a number.
Do not round too early. Keep several decimal places while solving. Round only when presenting a final answer. Also check interval endpoints carefully. A valid endpoint should be included when it satisfies the equation. Finally, verify unusual results in the original equation. This habit catches sign errors, unit mistakes, and values outside the requested domain.
Frequently Asked Questions
What equation form does this calculator solve?
It solves A × f(Bx + C) = D. The function can be sine, cosine, tangent, cotangent, secant, or cosecant. The tool reduces the equation, finds valid angle families, and converts them to x values inside your interval.
Which trigonometric functions are available?
You can select sine, cosine, tangent, cotangent, secant, or cosecant. Reciprocal functions use their related sine or cosine equation after the target is transformed.
How do degree and radian modes differ?
Degree mode treats phase and interval inputs as degrees. Radian mode treats them as radians. Use the unit already used by your source problem, and keep every angle value in that same unit.
What does coefficient A do?
Coefficient A scales the selected trigonometric function. The calculator divides D by A first. A cannot be zero because the equation would no longer contain a solvable trigonometric target.
What does coefficient B do?
Coefficient B changes the rate inside the angle expression. It affects the spacing of x solutions. A negative B is allowed, but zero is not allowed because x would not change the internal angle.
Why might no real solution appear?
Sine and cosine need reduced targets from −1 to 1. Secant and cosecant need magnitudes of at least 1. You can also have a valid general solution with no value inside your selected interval.
Why are there multiple answers?
Trigonometric functions repeat. One angle solution creates more solutions after full or half rotations. The calculator lists every matching x value that stays within the start and end values you provide.
Can I solve equations with more than one trigonometric function?
Rewrite or factor the equation first. Use identities to isolate one supported trigonometric expression equal to a number. Then solve each resulting factor or expression separately with the calculator.
Are interval endpoints included?
Yes. The calculator includes a start or end value when it satisfies the equation within normal floating-point tolerance. Verify endpoint results in the original equation when precision is especially important.
What does the residual value mean?
The residual is the absolute difference between the original left side and D after substitution. A value near zero indicates that the displayed solution satisfies the equation accurately.
How should I round the results?
Keep extra decimal places while working. Round only after checking the original equation. Accurate practice builds confidence and supports stronger mathematical decisions.