Problem 18 - Contemporary Precalculus

Question

Use your knowledge of special values to find the exact solutions of the equation. $$\tan x=1$$

Step-by-Step Solution

Answer

Answer: The exact solutions of the equation \(\tan x = 1\) are those values of x which are of the form \(\frac{\pi}{4} + n\pi\), where n can be any integer.

Step 1: Recognize the special angle for tangent function

We know that for some special angles, the tangent function has well-known exact values. In this case, we are looking for the angle where \(\tan x = 1\). The special angle that satisfies this equation is \(x = \frac{\pi}{4}\), because \(\tan \frac{\pi}{4} = 1\).

Step 2: Find the general solution for tangent function

The tangent function has a period of \(\pi\), which means it repeats its values every \(\pi\) radians. So, to find the general solution for this equation, we can add any integer multiple (n) of \(\pi\) to the special angle we found in step 1. The general solution for the equation \(\tan x = 1\) is: $$x = \frac{\pi}{4} + n\pi$$

Step 3: Write the exact solutions

Since we have found the general solution, we can write down the exact solutions for the equation \(\tan x = 1\): $$x = \frac{\pi}{4} + n\pi \quad \text{where} \, n \, \text{is any integer}$$ So, the exact solutions of the equation \(\tan x = 1\) are those values of x which are of the form \(\frac{\pi}{4} + n\pi\), where n can be any integer.

Key Concepts

Special Angles

Special angles in trigonometry are those angles for which the trigonometric functions take on simple, easy-to-remember values. These angles are typically found at 0, 30, 45, 60, and 90 degrees or their equivalents in radians, like 0, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\). For the tangent function, \(\tan x = 1\) when \(x\) is \(45\) degrees or \(\frac{\pi}{4}\) radians.

These angles are considered 'special' because they result in clean values that are easy to handle, like \(\sqrt{2}\), 1, or 0. They allow us to quickly solve trigonometric equations without a calculator. Knowing these angles is incredibly useful, especially in trigonometry problems where patterns and repeated values come into play.

Tangent Function

The tangent function, often denoted as \(\tan\), is one of the primary functions in trigonometry. It can be understood as the ratio of the sine function \(\sin\) to the cosine function \(\cos\).

For any angle \(x\), the tangent can be expressed as \(\tan x = \frac{\sin x}{\cos x}\). This relation implies that the tangent function is undefined where \(\cos x = 0\) because division by zero is undefined. In the unit circle, these are the angles where \(x = \frac{\pi}{2} + n\pi\).

One interesting property of the tangent function is that it repeats its values, or is periodic, every \(\pi\) radians. Thus, knowing the tangent of a special angle helps in finding its value over other intervals by adding integer multiples of \(\pi\). Additionally, the tangent function can take any real number, making it unique compared to sine and cosine which are bounded. Understanding these properties can simplify the process of solving trigonometric equations involving the tangent function.

General Solution of Trigonometric Equations

When we talk about the general solution of trigonometric equations, we refer to finding all possible angles that satisfy a given equation. For periodic functions like sine, cosine, and tangent, the solutions repeat at regular intervals.

For the tangent function, which is crucial in our exercise, the period is \(\pi\) radians. This means if \(x\) is a solution, then \(x + n\pi\) is also a solution for any integer \(n\). This property allows us to write the general solution for equations involving the tangent function.

For instance, in the equation \(\tan x = 1\), the special angle solution is \(x = \frac{\pi}{4}\). To express the general solution, we add \(n\pi\) to this angle, resulting in \(x = \frac{\pi}{4} + n\pi\), where \(n\) is any integer. This accounts for all angles where the tangent is equal to 1, including angles beyond a single revolution around the unit circle. Solving trigonometric equations generally involves determining one solution and then using the periodicity properties of the function to describe all solutions.