🔢 Trigonometric Equation Solver
Solve trigonometric equations step by step with detailed explanations. Find all solutions or just the principal value for basic trig equations.
Understanding Trigonometric Equations
Trigonometric equations are equations containing trigonometric functions of unknown angles. Unlike regular algebraic equations, trigonometric equations often have multiple solutions due to the periodic nature of trigonometric functions. Solving these equations requires understanding the behavior of sine, cosine, and tangent functions across different quadrants.
The key to solving trigonometric equations is recognizing that functions like sine and cosine repeat their values in predictable patterns. For example, sin(30°) = sin(150°) = 0.5, and this pattern continues with period 360°. Understanding these patterns allows us to find all solutions within a given interval or identify the general solution formula.
Our solver handles basic trigonometric equations of the form sin(x) = k, cos(x) = k, and tan(x) = k, providing both principal values and all solutions within one complete period. This foundation is essential for tackling more complex trigonometric equations in advanced mathematics.
Solve Trigonometric Equations
Solution Methods by Function
Solving sin(x) = k
Step 1: Check if |k| ≤ 1 (sine range)
Step 2: Find principal value: x₁ = arcsin(k)
Step 3: Find second solution: x₂ = π - arcsin(k)
Step 4: General solution: x = x₁ + 2πn or x = x₂ + 2πn
Example: sin(x) = 0.5
x₁ = arcsin(0.5) = π/6 = 30°
x₂ = π - π/6 = 5π/6 = 150°
Solving cos(x) = k
Step 1: Check if |k| ≤ 1 (cosine range)
Step 2: Find principal value: x₁ = arccos(k)
Step 3: Find second solution: x₂ = 2π - arccos(k)
Step 4: General solution: x = ±arccos(k) + 2πn
Example: cos(x) = 0.5
x₁ = arccos(0.5) = π/3 = 60°
x₂ = 2π - π/3 = 5π/3 = 300°
Solving tan(x) = k
Step 1: No range restriction (tangent domain: all reals)
Step 2: Find principal value: x₁ = arctan(k)
Step 3: Use period π: x₂ = x₁ + π
Step 4: General solution: x = arctan(k) + πn
Example: tan(x) = 1
x₁ = arctan(1) = π/4 = 45°
x₂ = π/4 + π = 5π/4 = 225°
Common Trigonometric Values
| Angle | Degrees | sin | cos | tan |
|---|---|---|---|---|
| 0 | 0° | 0 | 1 | 0 |
| π/6 | 30° | 1/2 | √3/2 | √3/3 |
| π/4 | 45° | √2/2 | √2/2 | 1 |
| π/3 | 60° | √3/2 | 1/2 | √3 |
| π/2 | 90° | 1 | 0 | undefined |
| π | 180° | 0 | -1 | 0 |
Frequently Asked Questions
Why do trigonometric equations have multiple solutions?
Trigonometric functions are periodic, meaning they repeat their values at regular intervals. For example, sin(30°) = sin(150°) = sin(390°) = 0.5. This periodicity means that once you find one solution, you can add multiples of the period to find infinitely many solutions.
What's the difference between principal value and general solution?
The principal value is the solution within the standard range of the inverse function ([-π/2, π/2] for arcsin and arctan, [0, π] for arccos). The general solution includes all possible solutions by adding appropriate multiples of the period to account for the function's periodicity.
How do I know if a trigonometric equation has solutions?
For sin(x) = k and cos(x) = k, solutions exist only if |k| ≤ 1, since the range of sine and cosine is [-1, 1]. For tan(x) = k, solutions always exist since tangent can take any real value. Always check the domain and range of the functions involved.