Problem 7 - Contemporary Precalculus

Question

Find all solutions of the equation. $$\cot x=2.3$$

Step-by-Step Solution

Answer

The general formula for all the solutions of the equation \(\cot(x) = 2.3\) is $$x = 0.4055 + n\pi$$, where \(n\) can be any integer.

Step 1: Find the initial solution within one period

To find the initial solution, we simply need to calculate the arccotangent of 2.3. The arccotangent is the inverse function of the cotangent, which gives us the angle for the given cotangent value. $$x = \arccot (2.3)$$

Step 2: Calculate the arccotangent value

We know that the cotangent function is defined as the ratio of the adjacent side to the opposite side in a right triangle, or $$\cot(x) = \frac{\cos(x)}{\sin(x)}$$. To find the inverse function, we can use the arctangent function. Since the arccotangent is the inverse of the cotangent, we have the following relationship: $$\cot(\arccot (x)) = x$$. By using the property of the arctangent (or tangent) function, we can rewrite the given equation as follows: $$x = \arctan(\frac{1}{2.3})$$

Step 3: Calculate the arctangent value

Calculate the arctangent of the inverse of 2.3: $$x = \arctan(\frac{1}{2.3})$$ After calculation, we get: $$x \approx 0.4055$$ radians

Step 4: Utilize cotangent periodicity

Since the cotangent function is periodic with a period of \(\pi\), we can find all solutions for the equation by adding integer multiples of \(\pi\) to the initial solution we found. So, the general formula for all the solutions is: $$x = 0.4055 + n\pi$$ where \(n\) can be any integer.

Key Concepts

Cotangent Function

The cotangent function, often abbreviated as "cot," is a fundamental trigonometric function. It is defined as the reciprocal of the tangent function. In trigonometry, \[ \cot(x) = \frac{1}{\tan(x)} \]You can also express cotangent using sine and cosine, like this:\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]Cotangent measures the ratio of the adjacent side to the opposite side in a right triangle. It is important for solving equations where the cotangent is specified, as it helps in determining angles when the value of the cotangent is known. The graph of the cotangent function features vertical asymptotes at integer multiples of \(\pi\), where the sine of the angle equals zero. It differs from the tangent function, having peaks and troughs that occur exactly where tangent has zero crossings.

Inverse Trigonometric Functions

Inverse trigonometric functions are utilized when you want to find an angle given a trigonometric ratio. They "undo" the trigonometric functions. For the cotangent function, its inverse is known as arccotangent, written as \( \arccot(x) \). The arccotangent function provides the angle whose cotangent is the specified value.

• \( \arccot(x) \): This is used when \( \cot(\theta) = x \) and you want to find \( \theta \).

To solve trigonometric equations like \( \cot x = 2.3 \), you first find the arccotangent of 2.3.A useful relationship is \(\arccot(x) = \arctan(\frac{1}{x})\)This transformation allows you to use a calculator's arctan function to find \( \arccot \). For the initial solution of \( \cot x = 2.3 \), it becomes \( \arctan(\frac{1}{2.3}) \approx 0.4055 \) radians. This is the reference angle corresponding to the given cotangent value.

Periodic Functions

Trigonometric functions often exhibit periodic behavior, meaning they repeat their values in regular intervals. The cotangent function, like the tangent, has a period of \( \pi \). This means that its values recur every \( \pi \) radians.

• Period of \( \cot(x) \): The function repeats every \( \pi \) radians, resulting in identical function values for inputs separated by \( \pi \).

When solving the equation \( \cot x = 2.3 \), after finding the initial solution \( x \approx 0.4055 \), you use this periodic property to generate all possible solutions:\[x = 0.4055 + n\pi\]where \( n \) is any integer. This accounts for the periodic nature of the function and ensures that all angles giving the same cotangent value are considered. Understanding periodicity is crucial for dealing with trigonometric equations, as it helps find a comprehensive set of solutions.