Problem 20 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = \frac{4\pi}{3} + 2n\pi\) and \(x = \frac{5\pi}{3} + 2n\pi\), where \(n\) is an integer.

Step 1: Isolate the sine function

First, we bring \(\sqrt{3}\) to the other side of the equation by subtracting \(\sqrt{3}\) from both sides, which leads to the equation:\[2\sin x = -\sqrt{3}\]Then, we divide both sides by 2 to isolate \(\sin x\):\[\sin x = -\frac{\sqrt{3}}{2}\]

Step 2: Identify the angle

We know that \(\sin(\frac{4\pi}{3}) = -\frac{\sqrt{3}}{2}\), thus \(x = \frac{4\pi}{3}\). It is also necessary to find the co-terminal angles. This is done by adding or subtracting multiples of one complete cycle which is \(2\pi\). Hence, the solutions will be in the form \(x = \frac{4\pi}{3} + 2n\pi\). But, \(\sin( \frac{5\pi}{3}) = - \frac{\sqrt{3}}{2}\) is another solution. Therefore, the solutions will be also in the form \(x = \frac{5\pi}{3} + 2n\pi\).

Step 3: Write down all solutions

The solutions for the equation \(2\sin x+\sqrt{3}=0\) will be:\[x = \frac{4\pi}{3} + 2n\pi \text{ and } x = \frac{5\pi}{3} + 2n\pi\] where \(n\) is an integer.

Key Concepts

Understanding the Sine Function

The sine function is a well-known function in trigonometry that relates an angle in a right triangle to the ratio of the length of the side opposite that angle to the hypotenuse. In the context of the unit circle, the sine of an angle is the y-coordinate of the point where the angle's terminal side intersects the circle.
It's important to understand that the sine function is periodic, repeating every \(2\pi\) radians (or 360 degrees). This periodicity means that the sine function will have the same value at multiple points along the angle's cycle, making it crucial for solving trigonometric equations like \(2\sin x + \sqrt{3} = 0\).
One more thing to note is the range of the sine function. It always lies between -1 and 1, inclusive. This means any solution for \(\sin x\) must fall within this range, highlighting the critical step of isolating the sine function to find its specific values.

Angle Solutions in Trigonometric Equations

When solving trigonometric equations, it's not only about finding one solution but rather all possible solutions, often called the general solution. For an equation like \(\sin x = -\frac{\sqrt{3}}{2}\), we first find a particular solution, such as \(x = \frac{4\pi}{3}\).
However, since sine functions are periodic, each angle has co-terminal angles that produce the same sine value, attained by adding multiples of a complete cycle \(2\pi\). Therefore, angle solutions for \(x\) are generally expressed in the form:

• \(x = x_0 + 2n\pi\)

where \(n\) is any integer and \(x_0\) is a specific solution within one full cycle.
Considering all solutions and angle possibilities ensures a comprehensive understanding of how trigonometric equations behave in their periodic nature.

The Role of Trigonometric Identities

Trigonometric identities are fundamental tools that help simplify and solve equations involving trigonometric functions. These identities include relationships like the Pythagorean identity, angle sum and difference identities, and double angle formulas.
In our context, the ability to transform the expression using identities helps us identify specific angles where the function takes particular values. For instance, identifying that \(\sin\left(\frac{4\pi}{3}\right) = -\frac{\sqrt{3}}{2}\) requires understanding how unit circle landmarks relate to sine values.
Moreover, trigonometric identities can make complex equations more approachable, guiding us towards patterns and relationships that aren't immediately evident. In solving techniques, they serve as a bridge between initially isolating the function and writing down all possible solutions, reinforcing the interconnectedness of trigonometric concepts.