Problem 38 - Precalculus Essentials

Question

Involve equations with multiple angles. Solve each equation on the interval \([0,2 \pi)\) $$\sin \left(2 x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}$$

Step-by-Step Solution

Answer

\(\frac{3\pi}{8}\)

Step 1: Write the given equation

The equation to be solved is \(\sin \left(2 x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}\). The right side of the equation is equal to \(\sin(\pi/4)\) or \(\sin(5\pi/4)\). Because the \(\sin\) function is positive in the first and second quadrants.

Step 2: Set the sine part of the equation equal to the angles where sine is equal to \(\sqrt{2}/2\)

This gives us two equations: \(2x-\pi/4=\pi/4\) and \(2x-\pi/4=5\pi/4\). We will solve each equation separately.

Step 3: Solve for \(x\) in each equation

For the first equation we get: \(2x=\pi/2 +\pi/4\), which simplifies to \(2x=3\pi/4\), and \(x=3\pi/8\). For the second equation we get: \(2x=5\pi/2+\pi/4\),which simplifies to \(2x=21\pi/8\) and \(x=21\pi/16\).

Step 4: Check the interval requirement for the solution

Now, all the solutions are to be in the interval [0, \(2\pi\)]. \(x=21\pi/16\) is larger than \(2\pi\), so it should be disallowed. So only the solution \(x=3\pi/8\) fits the interval.

Key Concepts

Understanding Multiple Angles in Trigonometry

When dealing with trigonometric equations involving multiple angles, it is crucial to recognize how they transform the standard functions. A multiple angle occurs when a trigonometric function involves expressions like \(2x\), \(3x\), or possibly \(2x - \pi/4\), as in this problem.
If you've encountered a sine equation like \( \sin(2x - \frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), it means that the sine function is being applied to an angle that is modified by a multiplier and possibly a phase shift. This can change the frequency and position of its wave.
Multiple angle identities help simplify and solve these kinds of equations. For instance, recognizing that the right side \( \frac{\sqrt{2}}{2} \) resembles the sine of a known angle, such as \( \pi/4 \) or \( 5\pi/4 \), can assist in simplifying the equation into standard sine equations by setting \( 2x - \pi/4 \) equal to these angles. This breaks the problem into discrete parts that are easier to solve, leveraging our understanding of sine’s behavior over standard intervals.

Exploring the Sine Function

The sine function is a fundamental concept in trigonometry, representing periodic oscillations. It oscillates between -1 and 1 with a period of \(2\pi\).
What sets it apart is its behavior within its critical quadrants: the sine function is positive in both the first (0 to \(\pi/2\)) and the second quadrant (\(\pi/2\) to \(\pi\)). Therefore, when solving the equation \( \sin(2x - \frac{\pi}{4}) = \frac{\sqrt{2}}{2} \), it suggests possible angles where sine value matches the given number.
This equates to \( \pi/4 \) and \( 5\pi/4 \) for the corresponding quadrants. Recognizing this pattern helps us set up our equations properly, thus leading to logical solutions. Having this insight also aids in understanding sine's cyclical nature, helping you forecast subsequent solutions if needed over different ranges.

Finding Interval Solutions in Trigonometric Equations

Interval solutions are about finding values of variables within a specific range that satisfy the equation. For trigonometry problems, such as these, the interval often references a section of the unit circle, like \([0, 2\pi)\), which means we focus on one full cycle of the trigonometric function.
For example, after solving the given multiple angle equation, not every mathematical solution might fit the interval requirement. It's essential to check: does the solution truly fall within \([0, 2\pi)\)? In our equation, \(x = 3\pi/8\) falls well within this range. However, a calculated value like \(x = 21\pi/16\) does not.
Additionally, understanding how periodicity can influence solutions means recognizing that solutions outside this interval might need converting back into our specified range, especially if they exceed \(2\pi\). This careful checking ensures correct results, reinforcing what the interval solution parameters demand.