Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation for solutions over the interval \([0,2 \pi)\) by first solving for the trigonometric finction. Do not use a calculator. $$2 \sin x+3=4$$

Step-by-Step Solution

Answer

\(x = \frac{\pi}{6} \text{ and } x = \frac{5\pi}{6}\).

Step 1: Isolate the Trigonometric Function

First, we need to isolate the term with the sine function on one side of the equation. The given equation is \(2 \sin x + 3 = 4\). To isolate \(2 \sin x\), subtract 3 from both sides:\[2 \sin x = 4 - 3\]This simplifies to:\[2 \sin x = 1\]

Step 2: Solve for Sine

To solve for \(\sin x\), divide both sides of the equation by 2:\[\sin x = \frac{1}{2}\]

Step 3: Find the General Solutions

Next, determine the values of \(x\) within the given interval \([0, 2\pi)\) for which \(\sin x = \frac{1}{2}\). Sine is \(\frac{1}{2}\) at two specific angles within this interval.The solutions are:\[x = \frac{\pi}{6} \quad \text{and} \quad x = \frac{5\pi}{6}\]These are because sine reaches \(\frac{1}{2}\) in the first and second quadrants.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, representing the y-coordinate of a point on the unit circle at a given angle. Understanding the sine function begins with its basic definition related to right triangles. In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. In the context of the unit circle, which has a radius of 1, this ratio translates to simply the y-value of a point defined by an angle. Thus, for any angle \(x\), the sine function returns the height of the corresponding point on this circle.

• The sine function is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.
• The values of sine range from -1 to 1, reflecting its position on the coordinate system.
• For example, at \(x = 0\), \(\sin x = 0\), and at \(x = \frac{\pi}{2}\), \(\sin x = 1\).

In the problem given, we isolate the sine term, \(\sin x = \frac{1}{2}\), which corresponds to specific angles where the y-coordinate equals a half. These angles can be linked back to specific known angles on the unit circle, providing exact values within the interval of interest.

Interval Notation

Interval notation is a way of representing a range of values, particularly useful for specifying certain subsets of the real number line in a concise format. When solving trigonometric equations like the one provided, it's important to express the solutions clearly within a specified range. The given problem specifies the interval \([0, 2\pi)\), meaning:

• The solutions must be greater than or equal to 0 but strictly less than \(2\pi\).
• \([\ ]\) indicates that the boundary number is included, while \((\ )\) means it is not.

This means we are looking for all possible angles corresponding to the trigonometric solution that remain within one full rotation, as defined by \(0\) to just under \(2\pi\) radians. Thus, once you solve for a trigonometric value, you must ensure that the extracted angle solutions fit within this specified range. In our particular equation, \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\) lie within this interval, fitting perfectly within its criteria.

Angle Measurement

Angle measurement in trigonometry is crucial for identifying how angles interact within circles and triangles. Angles are typically measured in degrees or radians, with radians being the primary unit in mathematical contexts, especially in trigonometry and calculus. One complete revolution around a circle is \(2\pi\) radians or 360 degrees. This equivalence is central to converting between degrees and radians.
Consider:

• \(\pi\) radians is half of a circle, equivalent to 180 degrees.
• For practical calculations in radians, angles are often multiples of \(\pi\), helping to simplify trigonometric problems.

In situating angle measurements like \(x = \frac{\pi}{6}\) or \(x = \frac{5\pi}{6}\) within a specific interval, it's understood that these angles correspond to specific positions on the unit circle. These positions help in understanding where \(\sin x\) takes certain values. Thus, recognizing radian measures highlights distinctive points in unit circle geometry, providing clarity on angle-specific properties and values in trigonometric equations.