Problem 6 - Contemporary Precalculus

Question

Find all solutions of the equation. $$\tan x=10$$

Step-by-Step Solution

Answer

Answer: The general solution for the equation $$\tan x = 10$$ can be expressed as $$x = 1.4711 + (n \cdot π)\,\text{radians}, \underline{\phantom{xxx}}where \underline{\phantom{xxx}} n \underline{\phantom{xxx}} is \underline{\phantom{xxx}} an \underline{\phantom{xxx}} integer.$$

Step 1: Find the principal solution

To find the principal solution of the equation, we will take the arctangent (or inverse tangent) of both sides of the equation: $$x = \arctan(10)$$ Now, use a calculator to find the arctan of 10: $$x \approx 1.4711\,\text{radians}$$ (rounded to 4 decimal places) This means that the principal solution of the equation is approximately 1.4711 radians.

Step 2: Express the general solution using periodicity

As mentioned earlier, the tangent function has a period of π, so the general solution to the equation can be expressed as: $$x = 1.4711 + (n \cdot π)$$ where n is an integer. So the equation $$\tan x = 10$$ has infinitely many solutions, and they can be expressed in the form: $$x = 1.4711 + (n \cdot π)\,\text{radians}, \underline{\phantom{xxx}}where \underline{\phantom{xxx}} n \underline{\phantom{xxx}} is \underline{\phantom{xxx}} an \underline{\phantom{xxx}} integer.$$

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the functions that allow us to find angles when the values of trigonometric ratios are known. In the context of the equation \( \tan x = 10 \), we need to find an angle \( x \) such that the tangent of \( x \) is 10. To do this, we take the inverse tangent, also known as the arctangent, of 10. This gives us the angle \( x = \arctan(10) \). Using a calculator, we determine that \( x \approx 1.4711 \) radians. This value is the principal solution, representing the smallest non-negative angle with a tangent of 10.

The importance of inverse trigonometric functions lies in their ability to bridge between ratios and angles in trigonometry, being especially useful in calculus, geometry, and engineering applications.

Periodicity of Trigonometric Functions

Trigonometric functions have repeating patterns, known as periods, which allow these functions to take on the same value at regular intervals. The tangent function, for instance, repeats its values every \( \pi \) radians. This means that if \( \tan x = 10 \) for a particular \( x \), the same equation holds for \( x + n \cdot \pi \), where \( n \) is any integer.

Because of this periodic nature, when we find the angle \( x = 1.4711 \) as a solution to \( \tan x = 10 \), we know that \( x = 1.4711 + n\cdot\pi \) for integer \( n \) also satisfies the equation. Periodicity is a fundamental property that enables trigonometric equations to have multiple solutions and is crucial for solving these equations in various mathematical and real-world contexts.

• Period for sine and cosine: \( 2\pi \)
• Period for tangent: \( \pi \)

General Solution of Trigonometric Equations

The general solution of a trigonometric equation expresses all possible values of the variable that satisfy the equation. For the equation \( \tan x = 10 \), the principal solution is \( x = \arctan(10) \approx 1.4711 \), but due to the periodicity of the tangent function, we can generalize this to describe infinitely many solutions.

These solutions take the form \( x = 1.4711 + (n \cdot \pi) \), where \( n \) is an integer. This formula captures every possible solution by accounting for the periodic repetition of the tangent function at intervals of \( \pi \) radians.

General solutions are useful because they provide a comprehensive answer to trigonometric equations, indicating all the angles, not just the principal ones, that satisfy the given conditions. Such solutions are essential in engineering and physics, where multiple rotations or cycles might affect outcomes.