Trigonometric Equations Solver

• Can: Reduce common forms to sin(x)=a, cos(x)=a, or tan(x)=a, including sin(2x)=k, cos(2x)=k, sin(2x)=k·cos(x), and cos(2x)=k·sin(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: Read flexible spellings — sinx, sin(x)^2, (sin(x))^2, and coefficients without an explicit * (like 2pi or sin(2x)=cos(x)) are all understood, not just one exact notation.
• 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.

• Enter a trig equation using x (example: cos(2x)=sin(x)).
• Check "Parsed as" under the input — it shows exactly how the equation was read, so a typo is obvious right away.
• Choose degrees or radians, and a solution set: general solution or an interval.
• Click Solve to get solutions, a unit-circle diagram, and optional step-by-step.

• 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) — unless one of a, b is zero, in which case it solves the simple base equation directly instead of an unnecessary conversion.
• For cos(2x)=m·sin(x), substitutes cos(2x)=1−2sin²(x) and solves the resulting quadratic in sin(x) directly.
• Returns a general solution (integer k) or solutions in a standard interval, plus a unit-circle plot of the principal solutions.
• If an equation type isn’t supported yet, it shows a clear message (so it won’t give wrong math).

Example 1 — Solve cos(2x)=1/2

1. Recognize the base form cos(2x)=k with k=1/2.
2. Let u=2x. Compute the reference angle: α=arccos(1/2)=π/3.
3. Use cosine symmetry: u=±α+2πk, so x=±π/6+πk.

Example 2 — Solve sin²(x)=3/4 (written as sin(x)^2)

1. Take the square root of both sides: sin(x)=±√3/2.
2. Solve each branch separately using sine symmetry.
3. sin(x)=√3/2 gives x=π/3+2πk or x=2π/3+2πk.
4. sin(x)=−√3/2 gives x=−π/3+2πk or x=4π/3+2πk.

Example 3 — Solve cos(2x)=sin(x)

1. Use the identity cos(2x)=1−2sin²(x).
2. 1−2sin²(x)=sin(x) → 2sin²(x)+sin(x)−1=0, a quadratic in sin(x).
3. Solve with the quadratic formula: sin(x)=(−1±3)/4, giving sin(x)=1/2 or sin(x)=−1.
4. Solve each case: sin(x)=1/2 gives x=π/6+2πk or 5π/6+2πk; sin(x)=−1 gives x=3π/2+2πk.

Example 4 — Solve √3·sin(x)−cos(x)=1

1. Match the linear mix form a·sin(x)+b·cos(x)=c with a=√3, b=−1, c=1.
2. Convert to a single sine: R=√(a²+b²)=2 and φ=atan2(b,a)=−π/6.
3. Solve 2·sin(x−π/6)=1 → sin(x−π/6)=1/2.
4. Solve for x−π/6, then add π/6 back: x=π/3+2πk or x=π+2πk.