Trig Equation Solver
Pick a trig function and a value c. The solver returns the two principal solutions in [0, 2π) and the general solution that captures every periodic copy. The reciprocal functions csc, sec and cot are rewritten in terms of sin, cos and tan before solving.
Why an inverse function is not enough
A trigonometric equation almost never has one answer. Because the functions repeat forever, any solution has infinitely many copies spaced a period apart — and within a single period there is usually a second, unrelated solution as well.
This solver reports both layers. It gives the principal solutions inside one full turn, and the general solution that captures every copy of them with a single integer parameter.
Taking an inverse sine returns one angle, and that is only ever part of the answer. The inverse functions are defined on a restricted range so they can be functions at all, which discards every solution outside it — including the second in the same period.
The general solution is what restores them. Reporting it alongside the principal values makes explicit that the answer is a family, not a number, which is the point most often lost when these equations are solved by calculator alone.
How to use this calculator
- Choose the function Any of the six. The three reciprocals are rewritten in terms of their partners before solving, and that rewrite is reported.
- Enter the right-hand side The value the function must equal. Sine and cosine only reach values between −1 and 1, and anything outside is rejected with an explanation.
- Pick the output unit Degrees or radians. This affects only how the answers are written, not which angles they are.
- Read the general solution, not only the principal ones The principal values sit inside one turn; the general solution is the complete answer, with an integer parameter standing for every repetition.
How the solutions are found
A reciprocal function is rewritten first: cosecant becomes sine of the reciprocal value, secant becomes cosine, cotangent becomes tangent — with a zero right-hand side rejected for the first two, since no reciprocal can be zero. Cotangent equal to zero gets its own case, because the reciprocal would be infinite: it means cosine vanishes, at a quarter turn and again three quarters round.
For sine the inverse gives an angle in the first or fourth quadrant and the second solution is its reflection about a quarter turn. For cosine the inverse gives an angle in the upper half and the second is its negative. For tangent there is one solution per period, and the second principal value is half a turn further on.
Each solution is then reduced into the interval from zero to a full turn, sorted, and any near-duplicates removed — which happens when the two coincide, as they do when sine equals exactly 1 or cosine equals exactly −1.
The general solution is written from the raw inverse rather than from the reduced principal values, with the period appended: a full turn for sine and cosine, a half turn for tangent. That is why the general form for cosine can be stated compactly as plus-or-minus one angle rather than as two separate families.
What each input means
- func Function — form field “Function”
- One of the six. Reciprocals are converted before solving, so the underlying equation is always in sine, cosine or tangent.
- c Right-hand side — form field “Right-hand side c”
- The target value. Bounded to between −1 and 1 for sine and cosine; unrestricted for tangent.
- unit Output unit — form field “Output unit”
- Degrees or radians, affecting presentation only.
- k Integer parameter
- Stands for any whole number in the general solution. Each value of it names one repetition of a principal solution.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
Sine equal to a half
A positive value inside the range, giving two principal solutions symmetric about a quarter turn.
Inputs Function = sin, Right-hand side c = 0.5, Output unit = deg
- Equation sin(x) = 0.5
- Principal solutions 30°, 150° (in [0, 360°))
- General solution x = 30° + 360°·k or x = 150° + 360°·k (k ∈ ℤ)
Result x ∈ {30°, 150°}; x = 30° + 360°·k or x = 150° + 360°·k (k ∈ ℤ)
The two principal answers are thirty and one hundred and fifty degrees. The second is the first reflected about ninety degrees, which is where the second solution always comes from for sine — and it is exactly the one an inverse sine on its own would miss.
The general solution has two families, one per principal value, each repeating every full turn. Sine takes each value in its range twice per period, so two families is the norm.
Cosine equal to a negative value
A negative right-hand side, so both solutions lie in the left half of the turn.
Inputs Function = cos, Right-hand side c = -0.5, Output unit = deg
- Equation cos(x) = -0.5
- Principal solutions 120°, 240° (in [0, 360°))
- General solution x = ±120° + 360°·k (k ∈ ℤ)
Result x ∈ {120°, 240°}; x = ±120° + 360°·k (k ∈ ℤ)
The answers are one hundred and twenty and two hundred and forty degrees — symmetric about half a turn rather than a quarter, which is the difference between cosine's symmetry and sine's.
The general solution is written as plus-or-minus a single angle plus a full turn. That compact form works for cosine precisely because its two solutions are negatives of each other.
A reciprocal function
Cosecant equal to 2, which is rewritten as sine equal to a half before anything is solved.
Inputs Function = csc, Right-hand side c = 2, Output unit = deg
- Equation csc(x) = 2
- Reciprocal rewrite csc(x) = 2 → sin(x) = 1/2 = 0.5
- Principal solutions 30°, 150° (in [0, 360°))
- General solution x = 30° + 360°·k or x = 150° + 360°·k (k ∈ ℤ)
Result x ∈ {30°, 150°}; x = 30° + 360°·k or x = 150° + 360°·k (k ∈ ℤ)
The rewrite line shows the conversion explicitly, and the answers are then identical to the first example. Cosecant equal to 2 and sine equal to a half are the same equation written two ways.
This is why the range restriction on cosecant looks inverted: it can never take a value strictly between −1 and 1, because its reciprocal would then have to exceed 1, which sine cannot do.
Reading the result
Two solutions per period, except for tangent
Sine and cosine each hit every value in their range twice within one full turn, so two principal solutions is the normal case. Tangent takes every value once per half turn, so it has effectively one family repeating twice as often.
Reading the general solution
The integer parameter stands for any whole number, positive, negative or zero. Setting it to zero recovers the principal value, and every other choice names another angle where the equation holds.
When the two solutions coincide
At the extremes of the range — sine equal to 1 or −1, cosine equal to either — the two principal solutions merge into one. The solver removes the duplicate rather than reporting the same angle twice.
When you would use this
Solving for a phase or an angle
Any equation reducing to a trigonometric function equalling a constant has this shape, and the general solution says which angles are admissible once a range is imposed.
Finding where a wave reaches a level
A periodic quantity crossing a threshold does so repeatedly, and the general solution enumerates those crossings rather than reporting only the first.
Assumptions and limitations
What this calculator assumes
- The equation is a single trigonometric function of x equal to a constant.
- Reciprocal functions are converted to their partners before solving.
- Principal solutions are reported within one full turn starting from zero.
- The general solution uses an integer parameter and the function's own period.
Where it stops being the right tool
- One function and one constant: equations mixing several trigonometric terms are out of scope.
- The argument must be x alone, so an equation in twice the angle needs substituting first.
- Answers are decimals rather than exact multiples of π or exact surds.
Common mistakes
Reporting only the inverse function's answer
Why it happens. A calculator's inverse sine returns one angle, and it looks like a complete answer. It is one of infinitely many, and usually not even the only one in the first turn.
How to avoid it. Read both principal solutions and then the general form. The second principal value is the one an inverse function discards by design.
Forgetting the periodic parameter
Why it happens. The principal solutions are concrete numbers and the general solution looks like extra notation rather than part of the answer.
How to avoid it. Include it unless the question restricts the interval. Without it the answer describes two angles instead of infinitely many.
Expecting a solution outside the range
Why it happens. Sine equal to 2 looks like an ordinary equation, and nothing about the expression signals that it is unsatisfiable.
How to avoid it. Check the value against the function's range. Sine and cosine never leave the interval from −1 to 1, and the solver says so rather than returning a complex angle.
Frequently asked questions
Why are there two principal solutions for sine and cosine?
Because each takes every value in its range twice within one full turn. For sine the two angles are symmetric about a quarter turn; for cosine, about the horizontal axis. An inverse function returns only one of them.
What does the general solution mean?
That every angle obtained by adding a whole number of periods to a principal solution is also a solution. The integer parameter stands for that whole number, so one line describes infinitely many angles.
When is there no solution?
When the value lies outside the function's range. Sine and cosine are confined between −1 and 1, so any value beyond that is unreachable. Cosecant and secant inherit the opposite restriction through their reciprocals: they can never lie strictly between −1 and 1.
Why does tangent behave differently?
Because it repeats every half turn rather than every full one, and takes every real value exactly once per period. There is no range restriction and effectively one family of solutions rather than two.