Question

Solve the equation tan() = -3. Enter your answers as a comma-separated list. Let k be any integer. Round terms to two decimal places where appropriate.

Answer 1

To solve the equation tan(\theta) = -3, find the acute angle using the inverse tangent, place the angle in the second and fourth quadrants where tangent is negative, and then express the general solutions considering the period of the tangent function.

Step-by-step explanation:

The question asks to solve the equation tan(\theta) = -3. Since tangents have a period of \(\pi\) radians (or 180 degrees), the general solutions can be written as:

1. Identify the acute angle whose tangent is 3. This is done using the inverse tangent function: \(\alpha = \tan^{-1}(3)\).
2. Determine the angles where the tangent is negative. Tangent is negative in the second and fourth quadrants, so our reference angle \(\alpha\) can be placed in these quadrants to get our specific solutions.
3. Write the general solutions considering the period of the tangent function. For the second quadrant: \(\theta = \pi - \alpha + k\pi\), and for the fourth quadrant: \(\theta = -\alpha + k\pi\), where \(k\) is any integer.

Round \(\alpha\) to two decimal places where appropriate and include that in your final expression for \(\theta\).