Problem 13 - 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. $$\cos x \cot x=\cos x$$

Step-by-Step Solution

Answer

Solutions are \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{4}, \frac{5\pi}{4} \).

Step 1: Analyze the Equation

The given equation is \( \cos x \cot x = \cos x \). Start by recognizing that we have a common factor on both sides of the equation, \( \cos x \).

Step 2: Factor Out Common Terms

Factor out \( \cos x \) from both sides: \[ \cos x ( \cot x - 1 ) = 0 \]This results in two separate equations to solve. 1. \( \cos x = 0 \)2. \( \cot x - 1 = 0 \)

Step 3: Solve \( \cos x = 0 \)

To find the values of \( x \) where \( \cos x = 0 \) within the interval \([0, 2\pi)\). The solutions are:\[ x = \frac{\pi}{2}, \frac{3\pi}{2} \]

Step 4: Solve \( \cot x = 1 \)

The equation \( \cot x = 1 \) is equivalent to \( \frac{\cos x}{\sin x} = 1 \), or \( \cos x = \sin x \). This occurs when:\[ x = \frac{\pi}{4}, \frac{5\pi}{4} \]

Step 5: Combine All Solutions

Combine the solutions from both conditions over the interval \([0, 2\pi)\):\[ x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{4}, \frac{5\pi}{4} \]

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos x \), is one of the fundamental trigonometric functions. It represents the x-coordinate of a point on the unit circle as it sweeps counterclockwise from the positive x-axis by an angle \( x \). The cosine function varies between -1 and 1 for all real numbers.
- **Periodicity:** The cosine function is periodic with a period of \( 2\pi \), meaning that it repeats its values every \( 2\pi \).- **Key Values:** At \( x = 0 \) and \( x = 2\pi \), \( \cos x = 1 \). At \( x = \pi \), \( \cos x = -1 \). Also, it is zero at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).
Understanding these key points helps in solving trigonometric equations as they guide the solution within given intervals, such as \( [0, 2\pi) \), which was important in the problem we solved.

Cotangent Function

The cotangent function, symbolized as \( \cot x \), is the reciprocal of the tangent function. It is defined as\[\cot x = \frac{\cos x}{\sin x}\].
- **Undefined Points:** It is undefined wherever \( \sin x = 0 \), which occurs at integer multiples of \( \pi \), such as \( x = 0, \pi, 2\pi \), etc.- **Ranges and Key Properties:** Unlike cosine, the cotangent can take any real value, going to infinity as \( \sin x \) approaches 0.
In the given problem, finding points where \( \cot x = 1 \) required recognizing angles where both \( \cos x \) equals \( \sin x \), which are found at \( x = \frac{\pi}{4} \) and \( x = \frac{5\pi}{4} \) within the interval \( [0, 2\pi) \).

Interval Notation

Interval notation is a mathematical representation of a set of numbers that includes all numbers between a given pair of endpoints. It is a concise way to express a range or region on the real number line.
- **Closed and Open Intervals:** A closed interval, \([a, b]\), includes both endpoints, while an open interval, \((a, b)\), excludes them. A half-open interval, like \([0, 2\pi)\), includes 0 but not \(2\pi\).- **Applications:** In the discussed trigonometric problem, we were tasked to find solutions within \([0, 2\pi)\). This means we looked for all solutions between 0 and just before hitting \(2\pi\), reflecting how complete cycles are handled in trigonometric problems.

Problem Solving Steps

Solving trigonometric equations involves systematic steps to isolate trigonometric functions and find their solutions within specified intervals. Let's quickly review the problem-solving steps.

• **Identify Common Factors:** Recognize common trigonometric functions on both sides. Factor them out if possible to simplify the equation.
• **Separate and Solve:** Breaking into simpler equations can often make the process more straightforward, as seen with \( \cos x (\cot x - 1) = 0 \).
• **Find Solutions in the Interval:** Use the properties of trigonometric functions to find solutions within given intervals. This was done for both \( \cos x = 0 \) and \( \cot x = 1 \).
• **Combine Solutions:** Finally, combine solutions from different parts to find all values within the interval. Ensuring no value is missed is crucial for completeness.

These steps not only apply to the specific problem but also serve as a toolbox for tackling various trigonometric equations you might encounter.