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Chapter 4: Problem 98

Find all solutions to the equation \(2 \sin x=\sqrt{3}\). Use \(k\) to represent any integer.

Short Answer

Expert verified
x = \( \pi/3 + 2k\pi \) and \( x = 2\pi/3 + 2k\pi \), where \( k \) is any integer.

Step by step solution

01

Isolate \( \sin x \)

Start with the equation: \( 2 \sin x = \sqrt{3} \). To isolate \sin x\, divide both sides by 2: \( \sin x = \frac{\sqrt{3}}{2} \).
02

Identify possible angles

Recognize that \sin(\pi/3) = \frac{\sqrt{3}}{2}\. Therefore, x could be \( \pi/3 \) or \( 2\pi/3 \) since sine is positive in the first and second quadrants.
03

Introduce the general solution

To capture all solutions, include the periodic nature of sine. The general solutions are \( x = \pi/3 + 2k\pi \) and \( x = 2\pi/3 + 2k\pi \) where \( k\) is any integer.

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Key Concepts

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

Isolating Trigonometric Functions
When tackling trigonometric equations, the first step is often isolating the trigonometric function. This makes it easier to work with and solve the equation.

In our exercise, we start with the equation: \( 2 \sin x = \sqrt{3} \).
To isolate \( \sin x \), we need to get rid of the 2 multiplying it. This is done by dividing both sides of the equation by 2:
\( \sin x = \frac{\sqrt{3}}{2} \).

By isolating the trigonometric function, we simplify the equation and make the next steps more straightforward. This step is crucial because it provides a clear path to finding the possible angles that satisfy the equation.
Sine Function
The sine function, written as \(\sin x\), takes an angle \(x\) and returns the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. The sine's range is from -1 to 1.

For our problem, we have isolated \( \sin x\) and found it equal to \(\frac{\sqrt{3}}{2}\). To solve the equation, we need to recall key angles where the sine value is known.
\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
Therefore, \pi/3\ is one of the solutions.

Since the sine function is periodic, it has multiple solutions for a single input value. This brings us to the next crucial step of identifying all possible solutions.
General Solutions for Trigonometric Equations
Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For the sine function, this interval is \(2\pi\).

To capture all solutions for a trigonometric equation, we introduce the general solution. This considers the periodic nature and allows us to find all possible values of \(x\) that satisfy the equation.
The general solutions for our equation \( 2 \sin x = \sqrt{3} \) are:
  • The solutions close to \( \pi/3\): \( x = \pi/3 + 2k\pi \).
  • The solutions close to \( 2\pi/3 \): \( x = 2\pi/3 + 2k\pi \)
where \( k \) is any integer.
Introducing the general solution ensures that we find all possible angles that satisfy the trigonometric equation.
Positive Sine Values in Quadrants
To identify all possible angles that solve \( 2 \sin x = \sqrt{3} \), we must understand where the sine function is positive.

The sine function is positive in the first and second quadrants of the unit circle.
  • In the first quadrant: any angle \( x \) where \ 0 \leq x \leq\frac{ \pi}{2} \
  • In the second quadrant: any angle \( x \) where \ \frac{ \pi}{2} \leq x \leq\pi \

For our isolated equation, we recognize that \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\). This angle falls in the first quadrant. The next step is to find the corresponding angle in the second quadrant. This is \(2\pi/3\), which equals \( \pi - \frac{\pi}{3} \).
Therefore, the two specific angles we identified are \(\frac{\pi}{3}\) and \(2\pi/3\), and we use them to derive the general solution.

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Most popular questions from this chapter

Find the exact value of each expression without using a calculator or table. a. \(\arcsin (1 / 2)\) b. \(\cos ^{-1}(-1 / 2)\) c. \(\tan ^{-1}(-1)\) d. \(\sin (\pi / 3)\) e. \(\cos (-\pi / 2)\) f. \(\sin ^{-1}(-1)\)

Simplify \(\sin (-x) \cos (-x) \tan (-x)\)

Find all angles in the interval \(\left[0^{\circ}, 360^{\circ}\right]\) that satisfy each equation. Round approximations to the nearest tenth of a degree. $$ \sin \alpha=-0.244 $$

Find all real numbers in the interval \([0,2 \pi)\) that satisfy each equation. Round approximate answers to the nearest tenth. $$ 6 \sin ^{2} x-2 \cos x=5 $$

Find the exact value of each composition without using a calculator or table. $$ \tan (\arccos (1 / 2)) $$
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