Problem 28 - 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. $$2 \sqrt{3} \cos \frac{x}{2}=-3$$

Step-by-Step Solution

Answer

Exact solutions are \(x = \frac{5\pi}{3}\) and \(x = \frac{7\pi}{3}\) radians.

Step 1: Simplify the equation

The given equation is \(2 \sqrt{3} \cos \frac{x}{2} = -3\). First, simplify by isolating the cosine term: \(\cos \frac{x}{2} = \frac{-3}{2 \sqrt{3}}\).

Step 2: Simplify the fraction

Simplify \(\frac{-3}{2 \sqrt{3}}\) by rationalizing the denominator: \[ \cos \frac{x}{2} = \frac{-3}{2 \sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{-3\sqrt{3}}{6} = \frac{-\sqrt{3}}{2} \].

Step 3: Use the cosine identity

Recall that \( \cos 30^\circ = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \). Hence, \(\cos \frac{x}{2} = -\frac{\sqrt{3}}{2}\) corresponds to angles where the cosine is negative, such as \(150^\circ\) (or \(\frac{5\pi}{6}\)) and \(210^\circ\) (or \(\frac{7\pi}{6}\)).

Step 4: Solve for \(\frac{x}{2}\)

Set \(\frac{x}{2} = \) each angle found: so \(\frac{x}{2} = \frac{5 \pi}{6}\) or \(\frac{x}{2} = \frac{7 \pi}{6}\).

Step 5: Solve for \(x\)

Multiply both sides of the equations \( \frac{x}{2} = \frac{5 \pi}{6} \) and \( \frac{x}{2} = \frac{7 \pi}{6} \) by 2 to solve for \(x\). Thus, \(x = \frac{5\pi}{3}\) and \(x = \frac{7\pi}{3}\).

Step 6: Convert to degrees (optional)

Convert \(x = \frac{5\pi}{3}\) and \(x = \frac{7\pi}{3}\) to degrees. Use \(180^\circ = \pi\): \(x = 300^\circ\) and \(x = 420^\circ\). Reduce \(420^\circ\) to the smallest nonnegative angle by subtracting \(360^\circ\), yielding \(60^\circ\).

Key Concepts

Radians and Degrees

Radians and degrees are two units for measuring angles. Understanding these units is key to solving trigonometric equations and problems.

• Radians: A radian measures angles based on the radius of a circle. Specifically, if you take the radius of a circle and wrap it along the edge, the angle it creates at the center of the circle is 1 radian. One entire circle (360°) is equal to 2π radians. This means that 1 radian equals about 57.2958 degrees.
• Degrees: Degrees are more intuitive for many people, dividing a circle into 360 equal parts. Common angles that we may encounter include 30°, 45°, 60°, and 90°.

Angles can be converted between radians and degrees using the conversion formula:
\(1° = \frac{\pi}{180} \) radians and \(1 \text{ radian} = \frac{180}{\pi}°\).
In solving problems, like finding the angles from \( x = \frac{5\pi}{3} \) and \( x = \frac{7\pi}{3} \), converting between these units ensures comprehension and satisfies requirements in certain mathematical contexts.

Cosine Function

The cosine function is fundamental in trigonometry. It defines the relationship between the angle and the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

• Range and Domain: The cosine function can describe any angle, though it particularly oscillates between -1 and 1 inclusive. Its domain, being all real numbers, leads to periodic outcomes.
• Periodicity: The cosine function is periodic with a period of \(2\pi\). This means that every \(2\pi\) units, the graph and function values repeat.
• Common Angles and Values: At \(0°, 90°, 180°\), the cosine values are 1, 0, -1 respectively. Understanding these standard angles helps in deriving solutions, such as recognizing that \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\).

In the equation given, the relationship \(\cos \frac{x}{2} = -\frac{\sqrt{3}}{2}\) indicates we are looking for angles where cosine has this negative value. Common angles like \(150° (\frac{5\pi}{6})\) and \(210° (\frac{7\pi}{6})\), where cosine is negative, aid in this solution process.

Solving Algebraic Fractions

Algebraic fractions occur when we deal with trigonometric identities and equations as seen in the original exercise. Simplifying these fractions is a necessary step in the calculation process.

• Rationalizing the Denominator: This technique involves multiplying the numerator and the denominator by a term that will eliminate any radicals in the denominator. This step keeps the value equivalent while streamlining the fraction.
• Simplifying Techniques: After rationalizing, ensure the fraction is reduced to its simplest form. For instance, in \( \frac{-3}{2\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{-\sqrt{3}}{2} \), the radical was removed from the denominator and the fraction was simplified.

Algebraic manipulation in trigonometric equations often reveals new identities or helps isolate terms, like the cosine component in our equation. Mastering these skills strengthens your ability to find angle solutions efficiently and accurately.