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Chapter 6: Problem 44

Find all real numbers that satisfy each equation. $$\tan (\pi x / 4)=1$$

Short Answer

Expert verified
The solution is \( x = 1 + 4k \), where \( k \) is an integer.

Step by step solution

01

- Understand the Equation

The equation given is \(\tan (\frac{\pi x}{4}) = 1\). Recall that the tangent function \(\tan \theta\) is equal to 1 at specific angles.
02

- Identify the Principal Value

The value of \(\theta\) for which \(\tan \theta = 1\) is \(\frac{\pi}{4} + k\pi\), where \(k\) is an integer because the tangent function repeats every \(\pi\) radians.
03

- Express the Angle in Terms of \(x\)

Since we have \(\frac{\pi x}{4}\) in our equation, set \(\frac{\pi x}{4} = \frac{\pi}{4} + k\pi\). Solve for \(x\) by isolating it on one side.
04

- Solve for \(x\)

Multiply both sides of \(\frac{\pi x}{4} = \frac{\pi}{4} + k\pi\) by 4 to get \(\pi x = \pi + 4k\pi\). Divide both sides by \(\pi\) to find \(x\). This results in \(x = 1 + 4k\).
05

- Provide the Solution

The set of all real numbers \(x\) that satisfy the equation is \(\{ x | x = 1 + 4k, k \in \mathbb{Z} \}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangent Function
The tangent function, noted as \(\tan{\theta}\), is one of the six fundamental trigonometric functions. It is particularly useful for looking at the ratio of the opposite side to the adjacent side in a right triangle. The tangent of an angle \(\theta\), written as \(\tan{\theta}\), can be derived from the sine and cosine functions: \(\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}\).
The tangent function has a few key characteristics:
  • It is undefined for angles where the cosine function is zero, as division by zero is not possible.
  • It has vertical asymptotes at these points (e.g., at \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \) and so on).
  • Its values range from negative to positive infinity, unlike the sine and cosine functions, which are bounded between -1 and 1.
Understanding the tangent function is crucial when solving trigonometric equations, as it helps to know when and how the function repeats its values.
Periodicity
The concept of periodicity in trigonometric functions is vital, especially for functions like \(\tan{\theta}\). Periodicity means that the function values repeat at regular intervals. The period of the tangent function is \(\pi\) radians. This means:
  • For any angle \(\theta\), the function \(\tan{\theta} = \tan{(\theta + k\pi)}\) where k is an integer.
  • The function repeats its pattern every \(\pi\) radians, unlike sine and cosine, which have a period of \(2\pi\).
This property helps us solve the equation \(\tan{(\frac{\pi x}{4})} = 1\). By knowing the periodic nature, we can understand that if \(\tan{\theta} = 1\) at an angle \(\frac{\pi}{4}\), it will also be 1 at every angle of the form \(\frac{\pi}{4} + k\pi\). Thus, we use this to find all possible values of \(\theta\) that satisfy the given equation.
Solution Set
A solution set in mathematics includes all values that satisfy a given equation. For the trigonometric equation \(\tan{(\frac{\pi x}{4})} = 1\), our goal is to find all values of \(x\) that make the equation true.
From the periodicity of the tangent function, we know that:
  • The tangent function equals 1 at angles of the form \(\frac{\pi}{4} + k\pi\), where k is any integer.
Given \(\frac{\pi x}{4} = \frac{\pi}{4} + k\pi \), we can isolate \(x\) by following these steps:
  • First, clear the fraction by multiplying by 4: \(\pi x = \pi + 4k\pi\).
  • Then, divide both sides by \(\pi\): \(x = 1 + 4k\).
Therefore, the solution set includes all real numbers of the form \(x = 1 + 4k\), where k is an integer, notated as \(\{ x | x = 1 + 4k, k \in \mathbb{Z} \}\). This shows that the equation has infinitely many solutions, all regularly spaced due to the periodic nature of the tangent function.

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Most popular questions from this chapter

Verify that each equation is an identity. $$\frac{\sin 4 t}{4}=\cos ^{3} t \sin t-\sin ^{3} t \cos t$$

Solve each equation. (These equations are types that will arise in Chapter 7.) $$\frac{\sin \alpha}{23.4}=\frac{\sin 67.2^{\circ}}{25.9} \text { for } 0^{\circ}<\alpha<90^{\circ}$$

Find all values of \(\alpha\) in degrees that satisfy each equation. Round approximate answers to the nearest tenth of a degree. $$\cos 2 \alpha=-0.22$$

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\) $$\sin (2 \alpha)=-8 / 17 \text { and } 180^{\circ}<2 \alpha<270^{\circ}$$

Find the amplitude, period, phase shift, and range for the function \(f(x)=5 \cos (2 x-\pi)+3\)
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