Problem 101 - Precalculus Essentials

Question

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$3 \cos x-6 \sqrt{3}=\cos x-5 \sqrt{3}$$

Step-by-Step Solution

Answer

The solutions for the equation \(3 \cos x-6 \sqrt{3}=\cos x-5 \sqrt{3}\) within the interval \([0,2\pi)\) are \(\frac{\pi}{6}\) and \(\frac{11\pi}{6}\).

Step 1: Isolate the Cosine Function

Firstly, subtract the \(\cos x\) term from both sides of the equation: \(3\cos x - \cos x = 6\sqrt{3} - 5\sqrt{3}\), which simplifies to \(2\cos x = \sqrt{3}\)

Step 2: Solve for x

Now, divide both sides by 2, resulting in \(\cos x = \frac{\sqrt{3}}{2}\). This equation states that the cosine of an angle x is \(\frac{\sqrt{3}}{2}\), and the solutions will be any angles on the unit circle that satisfy this condition.

Step 3: Identify the angles on a unit circle

The angles \(\frac{\pi}{6}\) and \(\frac{11\pi}{6}\) have a cosine value of \(\frac{\sqrt{3}}{2}\) on the unit circle. Since these angles lie in the range of \([0,2\pi)\), these are the solutions to the given equation.

Step 4: Check the solution

Plug these values back into the original equation to ensure they are indeed solutions.

Key Concepts

Understanding the Unit Circle

The unit circle is a crucial tool in trigonometry, essential for solving equations involving trigonometric functions like cosine and sine. It is a circle with a radius of 1, centered at the origin (0,0) on a coordinate plane. Every point on this circle corresponds to an angle measured from the positive x-axis.

• The x-coordinate of any point on the unit circle is the cosine of the angle.
• The y-coordinate is the sine of the angle.
• Angles are typically expressed in radians, where one full rotation around the circle equals \(2\pi\) radians.

By using the unit circle, we can find exact values for trigonometric functions. For example, common angles like \(\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}\), and their respective reflections, have known cosine and sine values that are often memorized to solve equations quickly.

Exploring the Cosine Function

The cosine function is one of the fundamental trigonometric functions, describing the relationship between an angle and the x-coordinate of the corresponding point on the unit circle.

• The cosine of an angle \(x\) is equal to the x-coordinate of the point on the unit circle at that angle.
• Cosine values range from -1 to 1.
• Cosine is an even function: \(\cos(-x) = \cos(x)\).

In the equation provided, \(\cos x = \frac{\sqrt{3}}{2}\), you're looking for angles where this relationship holds. This value points to specific, well-known angles on the unit circle where the x-coordinate equals \(\frac{\sqrt{3}}{2}\). These angles can be quickly identified using memorized unit circle values.

Finding Exact Values

Exact values in trigonometry refer to specific usually simple numbers that describe the result of a trigonometric function without approximations. They stem from the special angles and their known positions on the unit circle.

• For example, \(\cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\) and \(\cos(\frac{11\pi}{6}) = \frac{\sqrt{3}}{2}\).
• These values are easily remembered and are crucial for solving equations accurately.

Using exact values helps in maintaining precision, especially important when these results are used in subsequent calculations. When solving the equation \(\cos x = \frac{\sqrt{3}}{2}\), identifying \(x\) using these exact values ensures accuracy.

Using Interval Notation

Interval notation is a concise way of expressing the solution set of an equation, especially useful when dealing with functions like trigonometric equations. Instead of listing all possible solutions, interval notation succinctly represents a range of possible values.

• In this case, the problem is solved over the interval \([0, 2\pi)\).
• This notation means that the interval includes 0 but excludes \(2\pi\).
• It effectively translates to "all real numbers \(x\) such that \(0 \leq x < 2\pi\)."

Using interval notation allows for a clear, universal understanding of the range of valid solutions, helping avoid ambiguity and making communications in mathematics precise.