Problem 11 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The solutions to the equation are \( x = \frac{\pi}{3} + 2k\pi \) and \( x = \frac{2\pi}{3} + 2k\pi \), where \( k \) is any integer.

Step 1: Identify the value of x from the unit circle

From the unit circle, it is known that \( \sin x = \frac{\sqrt{3}}{2} \) when \( x = \frac{\pi}{3} \) or \( x = \frac{2\pi}{3} \) in the interval \([0, 2\pi]\).

Step 2: Consider the periodicity of the sine function

The sine function is periodic with a period of \( 2\pi \). Therefore, the equation will be true for \( x = \frac{\pi}{3} + 2k\pi \) and \( x = \frac{2\pi}{3} + 2k\pi \), where \( k \) is any integer. These are all the solutions to the equation.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is commonly used to determine the relationship between the sides of a right triangle and its angles. The sine of an angle in a right triangle is the ratio of the length of the side opposite the angle to the hypotenuse, the longest side.

For example, in a right triangle:

• If the angle is \( \theta \), "opposite" is the side opposite \( \theta \), and "hypotenuse" is the long side, then \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).

However, the sine function is also defined for all real numbers using the unit circle, allowing it to be applied beyond the context of right triangles.

Unit Circle

The unit circle is a crucial tool in trigonometry, helping to define the sine, cosine, and tangent functions for all real numbers. The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. Each angle on the unit circle corresponds to a point with coordinates \((x, y)\), where:

• The \( x \)-coordinate is \( \cos(\theta) \).
• The \( y \)-coordinate is \( \sin(\theta) \).

When dealing with the equation \(\sin x = \frac{\sqrt{3}}{2}\), we look for the angles on the unit circle where the \( y \)-coordinate is \(\frac{\sqrt{3}}{2}\).

These angles correspond to \( x = \frac{\pi}{3} \) and \( x = \frac{2\pi}{3} \) within the interval \([0, 2\pi]\). This understanding helps when solving trigonometric equations, especially when visualizing how the sine function behaves around a circle.

Periodicity

Periodicity is a fundamental property of trigonometric functions, meaning that they repeat their values at regular intervals. The sine function, for instance, is periodic with a period of \(2\pi\).

This means after every \(2\pi\) interval, the function repeats its values. For any angle \( x \), \( \sin(x) = \sin(x + 2k\pi) \) for any integer \( k \). This is why in our example, once we find specific solutions such as \( x = \frac{\pi}{3} \) and \( x = \frac{2\pi}{3} \), we can describe the complete solution set using these base solutions offset by multiples of the period:

• \( x = \frac{\pi}{3} + 2k\pi \)
• \( x = \frac{2\pi}{3} + 2k\pi \)

Thus, periodicity allows these solutions to cover all possible solutions for \( \sin x = \frac{\sqrt{3}}{2} \). This concept is vital in trigonometry to understand how and why trigonometric functions repeat and can help in graphing and solving related equations.