Problem 20 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find all values of \(x\) in the interval \([0,2 \pi]\) such that $$ \sin (4 x)=\frac{1}{\sqrt{2}} $$

Step-by-Step Solution

Answer

The solution for the given equation are: \(x = \pi/16, 3\pi/16, 5\pi/16, 7\pi/16, 9\pi/16, 11\pi/16, 13\pi/16, 15\pi/16\).

Step 1: Identify Corresponding Angles

We know that \(\sin(\pi/4) = \sin(3\pi/4) = 1/\sqrt{2}\). So, we want to find values of \(x\) such that \(4x\) equals to these angles. The general formula for that is \(4x = n\pi + (-1)^n \alpha\), where \(n\) is an integer and \(\alpha\) is the angle.

Step 2: Solve for x

To find \(x\), we divide both sides of the equation by 4. So, we get \(x = n\pi/4 + (-1)^n \alpha/4\).

Step 3: Substitute angle values

Substitute \(\alpha\) with \(\pi/4\) and \(3\pi/4\). Loop \(n\) from 0 to 7 to get the values of \(x\) in the interval \([0, 2\pi]\).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is often denoted as \( \sin(x) \) and is used to describe the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. It's periodic, which means it repeats its values in a regular pattern over intervals.

• Amplitude: This is the peak value of \(\sin(x)\), which ranges from -1 to 1.
• Period: The sine function has a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.
• Symmetry: It is an odd function, i.e., \( \sin(-x) = -\sin(x) \).

In our exercise, we identified that \( \sin(4x) = \frac{1}{\sqrt{2}} \). This suggests that the angle \(4x\) is hitting points on the sine wave where the function equals \( \frac{1}{\sqrt{2}} \). Understanding the properties of sine helps solve the equation with periodic solutions.

Angle Solutions

In trigonometry, finding angle solutions means determining the specific angles that satisfy a trigonometric equation. For the sine equation \( \sin(4x) = \frac{1}{\sqrt{2}} \), we look for angles that make this true.
Typically, the angles are associated with common sine values:

• \( \sin(\pi/4) = \frac{1}{\sqrt{2}} \)
• \( \sin(3\pi/4) = \frac{1}{\sqrt{2}} \)

Thus, about solving \(4x\), we use these angle solutions to express\( 4x \). These angles are critical milestones. They are generated from the known angle solutions that satisfy the function \( \sin(x) = \frac{1}{\sqrt{2}} \) within a specific interval. By setting \(4x = \pi/4 + 2n\pi\) and \(4x = 3\pi/4 + 2n\pi\), where \(n\) is an integer, we can derive actual values for \(x\).

General Form of Solutions

Mathematicians use a general formula to express solutions for trigonometric equations. This formula accounts for all possible solutions within given constraints, using periodicity.
In this case, the equation \(4x = n\pi + (-1)^n \alpha\) provides a generalized expression. \(n\) is an integer, and \(\alpha\) denotes angles that led to the specific sine value. The expression takes advantage of the sine function's periodic nature.

• \( n\pi \) ensures peaks or troughs of the sine's wave are matched.
• The term \((-1)^n\alpha\) selectively gives "+" or "-" versions required to account for symmetry.

Hence for our case, we substitute \(\alpha\) with \(\pi/4\) and \(3\pi/4\). Solving iteratively while obeying the constraints (\([0, 2\pi]\)), these transformations provide us the comprehensive set of \(x\) values satisfying the original trigonometric equation.