Solve the equation tanx-25.5

General
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solve the equation tanx-25.5

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Solve the equation tan(x) = 25.5

Answer: To solve the equation \tan(x) = 25.5, we need to consider the properties and behavior of the tangent function. Below are the steps to find solutions for this equation:

  1. Understanding the Tangent Function:

    • The tangent function, \tan(x), is periodic with a period of \pi radians. This means that if x is a solution, then x + k\pi (where k is any integer) will also be a solution.

  2. Using the Inverse Tangent Function:

    • The inverse tangent function, \arctan, will help us find the principal value of x. We can write:
      x = \arctan(25.5)

  3. Calculating the Principal Value:

    • Using a calculator or mathematical software to find \arctan(25.5):

      x \approx \arctan(25.5) \approx 1.53169 \text{ radians}

    • However, the tangent function is periodic, and the general solution needs to account for this periodicity.

  4. Including All Periodic Solutions:

    • Therefore, the general solution is:
      x = \arctan(25.5) + k\pi
    • Here, k is any integer, so the solutions are:
      x \approx 1.53169 + k\pi

Summary: The specific solution to the equation \tan(x) = 25.5 is approximately 1.53169 radians, but the general solution, accounting for the periodic nature of the tangent function, is given by:

x = \arctan(25.5) + k\pi \quad \text{for all integers } k.

This means the solutions are:

x \approx 1.53169 + k\pi \text{ radians, where } k \text{ is any integer}.

By considering multiple values of k, you can generate different solutions within different ranges if needed.