Problem 20 - Algebra and Trigonometry

Question

Find all solutions of each equation. $$ 2 \sin x+\sqrt{3}=0 $$

Step-by-Step Solution

Answer

The solutions for the equation are \(x = \frac{4\pi}{3} + k2\pi\) or \(x = \frac{5\pi}{3} + k2\pi\), where \(k\) is any integer.

Step 1: Simplify the equation

We should start by subtracting \(\sqrt{3}\) from both sides to isolate the term containing \(x\). This will result in the equation \(2 \sin x = -\sqrt{3}\).

Step 2: Solve for sin(x)

The next step is to solve for \(\sin x\). We can do this by dividing both sides of the equation by 2. As a result, we get \(\sin x = -\frac{\sqrt{3}}{2}\).

Step 3: Find the principal solution

The angle whose sine is \(-\frac{\sqrt{3}}{2}\) is \(\frac{4\pi}{3}\) or \(\frac{5\pi}{3}\) in the interval from 0 to \(2\pi\) (these are the principal solutions). So, one possible value for \(x\) is either \(\frac{4\pi}{3}\) or \(\frac{5\pi}{3}\).

Step 4: Find all solutions

The sine function has a period of \(2\pi\), which means it repeats its values every \(2\pi\). Therefore, all other solutions of the equation can be obtained by adding multiples of \(2\pi\) to the principal solution. These solutions can be expressed as \(x = \frac{4\pi}{3} + k2\pi\) or \(x = \frac{5\pi}{3} + k2\pi\), where \(k\) is an integer.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry. It relates an angle in a right triangle to the ratio of the opposite side over the hypotenuse. This function is pivotal not only in geometry but also in periodic phenomena such as waves.

• The sine function is usually denoted as \( \sin(\theta) \).
• It takes values between -1 and 1 for all angles \( \theta \).
• It is an odd function, meaning \( \sin(-\theta) = -\sin(\theta) \).

In our problem, we encountered the equation \( \sin x = -\frac{\sqrt{3}}{2} \). This is a specific value where the sine function takes particular angles, that will be explored in the principal solution section.

Principal Solution

A principal solution in trigonometry refers to the most immediate solutions of a trigonometric equation within a specified interval, typically \([0, 2\pi)\).
This interval is often used because it covers all possible unique values of the sine and cosine functions.

• To find these solutions, we consider where the sine function reaches the needed value.
• For the function \( \sin x = -\frac{\sqrt{3}}{2} \), two principal solutions exist at \( x = \frac{4\pi}{3} \) and \( x = \frac{5\pi}{3} \), since these angles yield \( \sin x = -\frac{\sqrt{3}}{2} \).

These solutions correspond to positions on the unit circle where the sine value is exactly \(-\frac{\sqrt{3}}{2}\).
When solving trigonometric equations, discovering the principal solution is a fundamental step.

Trigonometric Periodicity

Trigonometric functions, including the sine function, are periodic, meaning they repeat their values in regular intervals.

• For the sine function, this period is \(2\pi\).
• This means that if \( \sin x = a \), it will also equal \(a\) at \(x + 2\pi\), \(x + 4\pi\), and so on.

In our solution, after determining the principal solutions, any additional solutions can be found using this property of periodicity.
For example, adding \(2k\pi\) to the principal solutions \( \frac{4\pi}{3} \) and \( \frac{5\pi}{3} \) will generate all possible solutions, where \(k\) is an integer.

Angle in Radians

Radians are a way of measuring angles based on the radius of a circle. This unit is closely tied to how rotations and trigonometric functions are naturally expressed.

• A full circle is \(2\pi\) radians, equivalent to 360 degrees.
• Radians provide a direct link between the length of a circle's arc and the angle it subtends at the center.

In our exercise, all angles were expressed in radians, such as \( \frac{4\pi}{3} \) and \( \frac{5\pi}{3} \).
Using radians makes it easier to understand the periodicity of the sine function, as seen in the principal and other solutions.
It is a convenient and natural method for calculations involving periodic phenomena and in higher mathematics.