Problem 52 - Precalculus: A Concise Course

Question

Use a graph to solve the equation on the interval \([-2 \pi, 2 \pi]\). $$ \cot x=1 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{3\pi}{4}, \frac{7\pi}{4}\).

Step 1: Plotting the function

Start by graphing the function \(y=\cot x\). This function has vertical asymptotes, at multiples of \(\pi\), where the tangent function equals zero and therefore the cotangent function is undefined. Just like a cosine function, cotangent function also repeats every \(\pi\), so it's sufficient to graph one period and then repeat that pattern.

Step 2: Figuring out intersection points

Next, plot the line \(y=1\) on the same plot. The solution to the equation \(\cot x = 1\) will be the x-values of the points where this line intersects the plotted cotangent function. Make sure you're only considering the points in the given interval \([-2\pi , 2\pi]\).

Step 3: Identifying solutions

Identify these intersection points and their corresponding x-values. Bear in mind that the values should satisfy the interval \([-2\pi , 2\pi]\).

Key Concepts

Cotangent Function

The cotangent function, often abbreviated as \text{cot}(x) or \text{cotan}(x), is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function. Therefore, for any angle \(x\), the cotangent of \(x\) is defined as $$\text{cot}(x) = \frac{1}{\text{tan}(x)} = \frac{\text{cos}(x)}{\text{sin}(x)}.$$
Because it is the ratio of the adjacent side to the opposite side in a right-angled triangle, the cotangent function can be thought of as how much horizontal distance (adjacent side) corresponds to a unit of vertical distance (opposite side).

Characteristics of the Cotangent Function:

• Periodicity: Cotangent repeats its values every \(\pi\) radians, which is its period.
• Asymptotes: The cotangent graph features vertical asymptotes at every integer multiple of \(\pi\) radians, where the function is undefined since \(\text{tan}(x)\) is zero and division by zero is not possible.
• Range: The range of cot(x) is all real numbers, which means its values can be any number from negative to positive infinity.
• Quadrants: The sign of cotangent is positive in the first and third quadrants where cosine and sine have the same sign, and it is negative in the second and fourth quadrants where they have opposite signs.

When \(\cot x=1\), it implies that the angle \(x\) has an equal amount of adjacent and opposite sides, which occurs at angles where the corresponding point on the unit circle is at a 45-degree angle to the axes, or more precisely at \(\frac{\pi}{4}\) and \(\frac{5\pi}{4}\) within our specified interval.

Graphing Trigonometric Functions

Graphing trigonometric functions is essential for solving equations visually and understanding their behavior. The key principle is to represent the variations of the function's magnitude and sign as the angle changes.

Key Steps in Graphing Cotangent

• Identify the period: \(\cot x\) has a period of \(\pi\), so the pattern repeats every \(\pi\) units.
• Mark the asymptotes: At multiples of \(\pi\), where \(\tan x\) equals zero, \(\cot x\) is undefined and you'll draw vertical dashed lines to represent these asymptotes.
• Sketch the curve: Between the asymptotes, \(\cot x\) starts from negative infinity, crosses through zero, and goes up to positive infinity, thus having a shape that is the mirror image of the tangent function.
• Label the scale: Mark the axis with appropriate scales to ensure the period and asymptotes are accurately represented.

When sketching \(y=\cot x\) to find where it intersects \(y=1\), you should look for where the cotangent curve crosses the horizontal line at \(y=1\). This will be at specific points that reflect the cotangent of the angle being equal to \(1\), which graphically manifests as intersections on the plot.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved variables. They are powerful tools for simplifying and solving trigonometric expressions and equations.

Common Trigonometric Identities

• Reciprocal Identities: These relate a function to its reciprocal, such as \(\cot(x) = \frac{1}{\tan(x)}\) or \(\tan(x) = \frac{1}{\cot(x)}\).
• Pythagorean Identities: These express the fundamental relationship between the sine, cosine, and tangent functions, such as \(\sin^2(x) + \cos^2(x) = 1\).
• Angle Sum and Difference Identities: These allow the calculation of the sine, cosine, and tangent of the sum or difference of two angles.
• Double and Half-Angle Identities: Used to find the values of trigonometric functions at twice or half of an angle, like \(\sin(2x) = 2\sin(x)\cos(x)\).

These identities are instrumental when solving equations like \(\cot x=1\), as they might allow us to transform the given equation into a more familiar form, often involving \(\sin x\) and \(\cos x\). Knowing that \(\cot x = \frac{\cos x}{\sin x}\), if \(\cot x\) is equal to 1, we have \(\cos x = \sin x\), which occurs when both sides of a right triangle (representing \(\cos\) and \(\sin\)) are equal, leading us to specific angles within a circle where this is true.