Problem 27 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation ( \(x\) in radians and \(\theta\) in degrees) for all exact solutions where appropriate. Round approximale values in radians to four decimal places and approximate values in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. $$2 \sqrt{3} \sin \frac{x}{2}=3$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{2\pi}{3} + 2n\pi\) and \(x = \frac{4\pi}{3} + 2n\pi\) in radians.

Step 1: Isolate the Sine Function

Start by isolating the sine function. Given the equation \(2 \sqrt{3} \sin \frac{x}{2} = 3\), divide both sides by \(2 \sqrt{3}\) to get:\[ \sin \frac{x}{2} = \frac{3}{2 \sqrt{3}}. \]

Step 2: Simplify the Expression

Simplify the expression \(\frac{3}{2 \sqrt{3}}\):\[ \frac{3}{2 \sqrt{3}} = \frac{3}{2} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{3\sqrt{3}}{6} = \frac{\sqrt{3}}{2}.\]Thus, the equation becomes \(\sin \frac{x}{2} = \frac{\sqrt{3}}{2}.\)

Step 3: Determine the Angles in Radians

The equation \(\sin \theta = \frac{\sqrt{3}}{2}\) has solutions \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{2\pi}{3}\) in the interval \([0, 2\pi]\). Since \(\frac{x}{2} = \theta\), then \(x\) is:- \(x = 2\cdot \frac{\pi}{3} = \frac{2\pi}{3}\),- \(x = 2\cdot \frac{2\pi}{3} = \frac{4\pi}{3}\).

Step 4: General Solutions in Radians

The general solutions in radians are based on the periodicity of the sine function, which repeats every \(2\pi\):- \(x = \frac{2\pi}{3} + 2n\pi\),- \(x = \frac{4\pi}{3} + 2n\pi\),where \(n\) is any integer.

Step 5: Convert Solutions to Degrees if Required

If solutions are needed in degrees, convert radians to degrees by multiplying by \(\frac{180}{\pi}\):- For \(x = \frac{2\pi}{3}\), the degree measure is \(x = 2 \cdot 60^\circ = 120^\circ\).- For \(x = \frac{4\pi}{3}\), the degree measure is \(x = 2 \cdot 120^\circ = 240^\circ\).The general solutions in degrees would be:- \(x = 120^\circ + 360^\circ n\),- \(x = 240^\circ + 360^\circ n\).

Key Concepts

Radians and Degrees

Understanding radians and degrees is crucial when working with trigonometric equations. In mathematics, angles can be measured in degrees or radians. Degrees are a more familiar unit of angle measurement; for example, a right angle is 90 degrees. Radians, however, are often used in higher-level math because they relate more directly to unit circles and identify angles in terms of \((\pi)\).

• There are 360 degrees in a full circle.
• The full circle in radians is \(2\pi\) radians.
• Therefore, \(180\) degrees is equivalent to \(\pi\) radians.

To convert an angle from radians to degrees, you multiply by \(\frac{180}{\pi}\). Conversely, to convert from degrees to radians, multiply by \(\frac{\pi}{180}\). Knowing how to switch between these two forms allows you to solve equations much more flexibly. When solving trigonometric equations, keeping track of the units is essential to ensure accurate calculations.

Sine Function

The sine function is a fundamental concept in trigonometry. It helps in determining the ratio of the opposite side to the hypotenuse in a right-angled triangle. In the context of equations, the sine function describes a smooth wave that oscillates between -1 and 1.

• Its graph is a continuous wave that repeats every \(2\pi\) radians.
• Sine reaches its maximum value of 1 at angles of \(\frac{\pi}{2}\), \(2\pi + \frac{\pi}{2}\), etc.
• It hits its minimum value of -1 at \(\frac{3\pi}{2}\), \(4\pi + \frac{3\pi}{2}\), etc.

In trigonometric equations like \(2 \sqrt{3} \sin \frac{x}{2} = 3\), the sine function can be isolated and solved to find values of \(x\). Simplifying the expression, as shown in the step-by-step solution, \(\sin \frac{x}{2} = \frac{\sqrt{3}}{2}\), you find that this occurs at angles like \(rac{\pi}{3}\) and \(rac{2\pi}{3}\). Knowing the properties and behavior of the sine wave greatly assists in identifying these angle solutions.

General Solutions

When solving trigonometric equations, general solutions are crucial as they provide all possible solutions for an angle. Trigonometric functions are periodic, meaning they repeat values in regular intervals. For the sine function, any solution repeats every \(2\pi\) radians.The general solution formula allows us to express these repeating solutions. For example:

• If \(x = \frac{2\pi}{3}\), all solutions can be written as \(x = \frac{2\pi}{3} + 2n\pi\), where \(n\) is an integer.
• If \(x = \frac{4\pi}{3}\), then \(x = \frac{4\pi}{3} + 2n\pi\), following the same principle.

In practical terms, to find solutions within a specific range, you can substitute different integer values for \(n\) to determine all possible angles.The exercise also allows for solutions to be expressed in degrees, offering the flexibility to cater to different problem requirements. Converting radians to degrees using \(\frac{180}{\pi}\) provides a comprehensive understanding of the solution set.