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Chapter 10: Problem 18

In Exercises \(1-18,\) find all of the exact solutions of the equation and then list those solutions which are in the interval \([0,2 \pi)\). $$ \sin ^{2}(x)=\frac{3}{4} $$

Short Answer

Expert verified
The solutions in the interval \([0, 2\pi)\) are \( x = \frac{\pi}{3}, \frac{5\pi}{3}, \) and \( \frac{4\pi}{3} \).

Step by step solution

01

Start with the Given Equation

We are given the equation \( \sin^2(x) = \frac{3}{4} \). Our goal is to find all values of \( x \) that satisfy this equation, and then identify which of those solutions fall within the interval \([0, 2\pi)\).
02

Solving for \( \sin(x) \)

To solve for \( \sin(x) \), take the square root of both sides of the equation. The square root of \( \sin^2(x) \) is \( |\sin(x)| \), therefore:\[ \sin(x) = \pm \sqrt{\frac{3}{4}} = \pm \frac{\sqrt{3}}{2} \]
03

Determine General Solutions

We know that \( \sin(x) = \frac{\sqrt{3}}{2} \) at angles where \( x = \frac{\pi}{3} + 2k\pi \) and \( \sin(x) = -\frac{\sqrt{3}}{2} \) at angles where \( x = \frac{4\pi}{3} + 2k\pi \) for any integer \( k \). These expressions represent the general solutions.
04

List Solutions in the Given Interval

Now, we restrict the solutions to the interval \([0, 2\pi)\). For \( \sin(x) = \frac{\sqrt{3}}{2} \), the valid solutions are \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \) (as these are within the first and second revolutions of the interval). Similarly, for \( \sin(x) = -\frac{\sqrt{3}}{2} \), \( x = \frac{4\pi}{3} \) is the acceptable solution within the interval.
05

Compile and Verify Solutions

Thus, the solutions in the interval \([0, 2\pi)\) are \( x = \frac{\pi}{3}, \frac{5\pi}{3}, \) and \( \frac{4\pi}{3} \). Verify by substituting back into the original equation to ensure that these values satisfy \( \sin^2(x) = \frac{3}{4} \).

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Key Concepts

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

Sine Function
The sine function is a fundamental concept in trigonometry, representing the ratio of the length of the opposite side to the hypotenuse in a right triangle. It is denoted as \( \sin(x) \), where \( x \) is the angle. This function oscillates between -1 and 1 as \( x \) ranges from \( 0 \) to \( 2\pi \) radians. This range shows the periodic nature of the sine wave, repeating every \( 2\pi \) radians.
A few key points about the sine function include:
  • It is an odd function, meaning \( \sin(-x) = -\sin(x) \).
  • Max value: 1 (occurs at \( \frac{\pi}{2} \), \( \frac{5\pi}{2} \), etc.).
  • Min value: -1 (occurs at \( \frac{3\pi}{2} \), \( \frac{7\pi}{2} \), etc.).
  • The zero crossings (where \( \sin(x) = 0 \)): \( 0, \pi, 2\pi \), and so on.
Understanding these properties aids in solving equations like \( \sin^2(x) = \frac{3}{4} \) since it involves finding values of \( x \) where sine is either positive or negative.
Exact Solutions
Finding exact solutions in trigonometry means identifying all possible angles \( x \) that satisfy a given equation, expressed in terms of \( \pi \). For instance, in the equation \( \sin^2(x) = \frac{3}{4} \), the task is to find all angles \( x \) whose sine value, either positive or negative, equals \( \frac{\sqrt{3}}{2} \).
To solve \( \sin(x) = \pm \frac{\sqrt{3}}{2} \), consider the sine values associated with common angles:
  • \( \sin(x) = \frac{\sqrt{3}}{2} \) occurs at angles \( x = \frac{\pi}{3} + 2k\pi \) and \( x = \frac{2\pi}{3} + 2k\pi \), where \( k \) is an integer.
  • \( \sin(x) = -\frac{\sqrt{3}}{2} \) appears at \( x = \frac{4\pi}{3} + 2k\pi \) and \( x = \frac{5\pi}{3} + 2k\pi \).
By identifying these general angles, we can narrow down the solutions that fit within any specified range or interval.
Interval Notation
Interval notation is a mathematical concept used to describe a set of numbers lying within two bounds. It elegantly informs us of the start and end of a range, including or excluding endpoints as necessary. In trigonometry, interval notation is crucial when restricting solutions to a specific domain like the interval \([0, 2\pi)\).
In the given problem with \( \sin^2(x) = \frac{3}{4} \), only those solutions that fall within the \([0, 2\pi)\) interval must be selected:
  • \([0, 2\pi)\) means the interval includes 0 but excludes \(2\pi\).
  • The acceptable solutions are those angles derived from both positive and negative sine values that exist within this boundary.
  • For this equation, these solutions are \( x = \frac{\pi}{3}, \frac{5\pi}{3}, \frac{4\pi}{3} \).
By using interval notation correctly, you can efficiently communicate which solutions are valid based on any mathematical constraints imposed by the domain of the function or equation.

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