All calculatorsTrigonometric Equations Solver
Solve common trigonometric equations for x (like sin(x)=1/2, 2cos²(x)−1=0, sin(2x)=√3·cos(x), or sin(x)+cos(x)=1). Choose degrees or radians, and get either a general solution (with integer k) or solutions in a standard interval.
Background
Trig equations often have infinitely many solutions because trig functions are periodic. For example, sin(x) repeats every 2π radians (or 360°). This solver focuses on popular homework forms that reduce using identities (like sin(2x)=2sin(x)cos(x)) and then solves base equations like sin(x)=a, cos(x)=a, or tan(x)=a.
What this solver can (and can’t) do
- Can: Reduce common forms to sin(x)=a, cos(x)=a, or tan(x)=a, including sin(2x)=k and sin(2x)=k·cos(x).
- Can: Solve linear mixed sums of the form a·sin(x)+b·cos(x)=c by converting to a single shifted sine (R·sin(x+φ)=c).
- Can’t: Fully arbitrary trig algebra (example: sin(x)+cos(2x)=1) or equations that need numeric root-finding. Unsupported types show a clear message instead of guessing.
- Tip: Use pi or π, sqrt( ) or √, and ^2 for squares.
How to use this solver
- Enter a trig equation using x (example: sin(2x)=√3·cos(x)).
- Choose degrees or radians.
- Choose a solution set: general solution or an interval.
- Click Solve to get solutions (and optional step-by-step).
How this solver works
- Recognizes supported patterns and reduces them using identities.
- For a·sin(x)+b·cos(x)=c, converts to R·sin(x+φ)=c (no root-finding).
- Returns a general solution (integer k) or solutions in a standard interval.
- If an equation type isn’t supported yet, it shows a clear message (so it won’t give wrong math).
Formula & Equation Used
Double-angle identity: sin(2x) = 2sin(x)cos(x)
Linear mix identity: a sin(x)+b cos(x)=R sin(x+φ), where R=√(a²+b²), φ=atan2(b,a)
Periodicity: sin(x + 2πk)=sin(x), cos(x + 2πk)=cos(x)
Example Problems & Step-by-Step Solutions
Example 1 — Solve sin(x)=1/2
- Recognize the base form sin(x)=a with a=1/2.
- Compute the reference angle: α=arcsin(1/2)=π/6.
- Use sine symmetry: x=α+2πk or x=(π−α)+2πk.
Example 2 — Solve sin(2x)=√3·cos(x)
- Use identity: sin(2x)=2sin(x)cos(x).
- 2sin(x)cos(x)=√3·cos(x) → cos(x)(2sin(x)−√3)=0.
- Case 1: cos(x)=0.
- Case 2: sin(x)=√3/2.
- Combine both families into the final solution set.
Example 3 — Solve sin(x)+cos(x)=1
- Match the linear mix form a·sin(x)+b·cos(x)=c with a=1, b=1, c=1.
- Convert to a single sine: R=√(a²+b²)=√2 and φ=atan2(b,a)=π/4.
- So sin(x)+cos(x)=√2·sin(x+π/4).
- Solve √2·sin(x+π/4)=1 → sin(x+π/4)=1/√2=√2/2.
- Solve the sine equation for x+π/4, then subtract π/4.
Frequently Asked Questions
Q: What mixed sums does it support?
It supports linear forms like a·sin(x)+b·cos(x)=c (including sin(x)+cos(x)=1). More complex sums like sin(x)+cos(2x)=1 are not supported yet in v1.
Q: Can I type √ and π?
Yes. You can use √ or sqrt( ), and π or pi. Example: sin(2x)=√3·cos(x) or cos(x)=pi/6.
Q: Why do I see “+ 2πk” or “+ 360°k”?
Trig functions are periodic. k is any integer, and adding a full rotation keeps the trig value the same.
Q: What if my equation isn’t supported?
The solver will show a clear “not supported yet” message instead of guessing.
