In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅).tan x = ﹣3

Verified step by step guidance

1. Identify the equation to solve: \( \tan x = -3 \).
2. Recognize that the tangent function is periodic with a period of \( \pi \).
3. Use the inverse tangent function to find the principal value: \( x = \tan^{-1}(-3) \).
4. Determine the general solution for \( x \) using the periodicity of the tangent function: \( x = \tan^{-1}(-3) + n\pi \), where \( n \) is an integer.
5. Find the specific solutions within the interval \([0, 2\pi)\) by substituting appropriate values of \( n \) and using a calculator to compute \( x \) to four decimal places.

Key Concepts

Tangent Function

The tangent function, denoted as tan(x), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It is periodic with a period of π, meaning it repeats its values every π radians. Understanding the behavior of the tangent function, including its asymptotes and where it is positive or negative, is crucial for solving equations involving tan(x).

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, are used to find angles when the value of a trigonometric function is known. For example, if tan(x) = -3, we can use arctan(-3) to find the reference angle. However, since the tangent function is periodic, we must consider all possible angles within the specified interval, which in this case is [0, 2π).

Calculator Usage for Trigonometric Equations

Using a calculator to solve trigonometric equations involves inputting the equation correctly and understanding the settings, such as whether the calculator is in degree or radian mode. For the equation tan(x) = -3, the calculator will provide a principal value, but additional solutions must be found by considering the periodic nature of the tangent function and the specified interval [0, 2π).