The general solution for the trigonometric equation involving tangent, specifically tan(x) = tan(α), is given by: x = nπ + α, where α lies within the interval (-π/2, π/2) and n is any integer (n ∈ Z).
Understanding the Tangent Solution
The tangent function has a period of π. This means that the value of tan(x) repeats every π units. Therefore, if tan(x) equals tan(α), then x can be equal to α plus any integer multiple of π.
Key Components:
- x: This represents the variable we are solving for.
- α: This is a known angle, which serves as our reference point. It must be within the interval (-π/2, π/2).
- n: This is any integer (..., -2, -1, 0, 1, 2, ...). It accounts for all the possible solutions across the periodic nature of the tangent function.
- π: This represents the mathematical constant approximately equal to 3.14159. It is used as the period of the tangent function.
How it Works:
Because tangent repeats every π radians, there are multiple values of x that would satisfy tan(x) = tan(α). The formula x = nπ + α effectively generates all these solutions.
Example:
Let’s say we want to solve tan(x) = tan(π/4). Here, α = π/4. The general solution would be:
- x = nπ + π/4
Here's how some solutions would look for different integer values of n:
| n | x = nπ + π/4 |
|---|---|
| -2 | -2π + π/4 = -7π/4 |
| -1 | -π + π/4 = -3π/4 |
| 0 | 0π + π/4 = π/4 |
| 1 | π + π/4 = 5π/4 |
| 2 | 2π + π/4 = 9π/4 |
Practical Insights
- Reference Angle: The value 'α' is often referred to as the reference angle. It's the specific angle that the tangent function is being compared to.
- Integer Multiples of π: The 'nπ' component accounts for the periodic behavior of the tangent function.
- Interval for α: Restricting the value of α to (-π/2, π/2) ensures that we work with the principal values of tangent function
Conclusion
In summary, to find all possible solutions for an equation in the form of tan(x) = tan(α), you should use the formula x = nπ + α where n is any integer and α is the reference angle. This formula correctly takes into account the periodicity of the tangent function.
