Problem 1 - Precalculus with Calculus Previews

Question

In Problems \(1-6\), find all solutions of the given trigonometric equation if \(x\) represents an angle measured in radians. $$ \sin x=\sqrt{3} / 2 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{\pi}{3} + 2n\pi\) and \(x = \frac{2\pi}{3} + 2n\pi\) for integer \(n\).

Step 1: Identify Angles with Given Sine Value

The sine of an angle is equal to \(\frac{\sqrt{3}}{2}\) at specific angles on the unit circle. Recall that \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\) and \(\sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2}\). These are the two angles in the first and second quadrants where the sine is positive and equals \(\frac{\sqrt{3}}{2}\).

Step 2: Express General Solution for Sin Equation

The sine function is periodic with a period of \(2\pi\). Therefore, the general solutions for the equation \(\sin x = \frac{\sqrt{3}}{2}\) can be expressed by considering the periodicity. Thus, the solutions are given by:- \(x = \frac{\pi}{3} + 2n\pi\)- \(x = \frac{2\pi}{3} + 2n\pi\)where \(n\) is any integer.

Key Concepts

Sine Function

The sine function, commonly denoted as \( \sin \theta \), is one of the fundamental trigonometric functions. It's especially useful for finding the height of a point on the unit circle. For any angle \( \theta \), the sine function gives the y-coordinate of the corresponding point on a unit circle:

• The maximum value the sine function can have is 1, and its minimum is -1. This range makes it crucial for modeling wave-like or oscillatory behavior.
• At angles like \( \frac{\pi}{3} \) or \( \frac{2\pi}{3} \), the sine function reaches specific values like \( \frac{\sqrt{3}}{2} \), often found using trigonometric tables or the unit circle.
• The function is periodic, meaning it repeats its pattern at regular intervals.

Understanding the sine function is essential for solving trigonometric equations where angles need precise measurement.

Unit Circle

The unit circle is a circle with a radius of one unit, centered at the origin of a coordinate plane. It is the foundation for defining trigonometric functions in terms of angle measures:

• The unit circle helps visualize and solve trigonometric equations such as sine, cosine, and tangent.
• Angles on the unit circle are usually measured in radians, where \( 2\pi \) radians correspond to a full circle (360 degrees).
• For specific angles, such as \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \), the unit circle helps easily identify the sine values, as these points lie at the y-coordinates \( \frac{\sqrt{3}}{2} \).

Using the unit circle, one can efficiently find all angle measures corresponding to a particular sine value.

General Solution

In trigonometry, finding the general solution of an equation involves capturing all possible angle measures (or solutions) that satisfy the equation. For the equation \( \sin x = \frac{\sqrt{3}}{2} \):

• The angles identified through the unit circle for this sine value are \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \).
• Because the sine function repeats every \( 2\pi \), we can express these solutions generally as:\( x = \frac{\pi}{3} + 2n\pi \) and \( x = \frac{2\pi}{3} + 2n\pi \).
• \( n \) represents any integer, ensuring the solutions include all rotations around the circle.

Understanding the general solution enables you to find all possible angles that meet the provided trigonometric condition.

Periodicity

Periodicity is a key characteristic of trigonometric functions, meaning they repeat their values in regular intervals. For the sine function:

• The period is \( 2\pi \), which means every \( 2\pi \) radians, the sine function returns to its original value.
• This feature helps to construct general solutions by adding multiples of \( 2\pi \) to the primary solutions, as seen in the given equation \( \sin x = \frac{\sqrt{3}}{2} \).
• Periodicity ensures that not only specific angles are solutions but all subsequent angles, repeated every \( 2\pi \), will also solve the equation.

Recognizing periodicity allows for predicting function behavior and solving complex trigonometric equations effectively.