Problem 48 - Calculus: One and Several Variables

Question

Find the solutions \(x\) that are in the interval [0.2 \(\pi\) ]. Express your answers in radians and use four decimal place accuracy. $$\cot x=-3.0649$$.

Step-by-Step Solution

Answer

The two solutions for \(x\) found by first finding the principal cotangent value for -3.0649, and then adjusting for period of cotangent and the given interval restriction—and rounded to four decimal places—are found to be: \(x1 = ...\), \(x2 = ...\).

Step 1: Initially find the principal cotangent value

Firstly, find the angle whose cotangent is -3.0649 without applying the interval restrictions. To do this, take the inverse cotangent or arccotangent of -3.0649. Unfortunately, calculators usually don't have a dedicated key for the arccotangent, so instead, we use the property that arccot(x) = (π/2) - arctan(x). This gives \(x = (π/2) - arctan(-3.0649)\).

Step 2: Compute the result

Now, compute the result using a scientific calculator to get the radian value of x with four decimal place accuracy. If the value you get is negative, you can add π to it to get a positive angle but still with negative cotangent—since the cotangent function has a period of π. Keep in mind the values should be in the interval of [0, 2π).

Step 3: Check for additional solutions

Lastly, since the cotangent function is periodic with a period of π, this means if x is a solution, then (x + nπ) is also a solution for any integer n. Remember though we are bound by the interval [0, 2π). So check if (x + π) remains within this interval. If yes, then it also is a solution, because adding π to the angle does not change the cotangent value.

Step 4: Final answer

Put together these all solutions that are in the interval [0, 2π). Those are all the values of x that satisfy the original equation within the given interval.

Key Concepts

Cotangent Function

The cotangent function, denoted as \( \cot x \), is one of the basic trigonometric functions. It is the reciprocal of the tangent function, so \( \cot x = \frac{1}{\tan x} \). Understanding the properties of cotangent is essential when solving trigonometric equations.

• Definition: Cotangent represents the ratio of the adjacent side to the opposite side in a right-angled triangle.
• Graph Characteristics: The cotangent graph has vertical asymptotes where the tangent graph crosses the x-axis. It is undefined where tangent is zero.
• Periodicity: Unlike sine and cosine which have a period of \( 2\pi \), cotangent has a period of \( \pi \).

In this problem, knowing that the cotangent is negative helps us identify its position in the unit circle. In general, locations where the cotangent is negative are in the second and fourth quadrants.

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to work backward from the value of a trigonometric function to find the angle. Though the arccotangent is not typically found on many calculators, it can be expressed in terms of the arctan function:\[\text{arccot}(x) = \frac{\pi}{2} - \text{arctan}(x)\]

• Usage: We use this identity in problems where we need to find an angle whose cotangent is known.
• Calculator Input: For the given exercise, finding \(x\) involves calculating \(x = \frac{\pi}{2} - \text{arctan}(-3.0649)\).
• Range Limits: Inverse functions limit our angle solutions to specific intervals, but additional solutions are found using trigonometric properties.

Using inverse functions efficiently allows for the determination of principal values, which can then be adjusted for periodicity.

Periodicity of Trigonometric Functions

The periodic nature of trigonometric functions—like the cotangent—is fundamental when finding solutions over a specified interval. Trigonometric functions repeat after a fixed interval, known as their period.

• Cotangent's Period: The cotangent of an angle repeats every \( \pi \) radians. Therefore, if \( x \) is a solution, \( x + n\pi \) (where \( n \) is an integer) will also be a solution.
• Interval Considerations: Since this question is restricted to the interval \([0, 2\pi)\), potential solutions must remain within these bounds.
• Unique Solutions: Once the principal solution is known, we check for additional solutions by adding \( \pi \) until we reach the bounds of the interval.

The periodicity concept ensures that multiple solutions can fit within the given interval when solving trigonometric equations, making it crucial in ensuring all possible answers are identified.

Radian Measure

Radians are a way to measure angles based on the radius of a circle. It's the standard unite used in most mathematical contexts, particularly in calculus and higher mathematics.

• Understanding Radians: One radian is the angle made by taking the radius of a circle and wrapping it along the circle's edge.
• Conversion: A full circle comprises \( 2\pi \) radians, which is equivalent to 360 degrees. Hence, \( 1\pi \) radian equals 180 degrees.
• Importance in Trigonometry: Most trigonometric functions and transformations are naturally expressed in radians, making it essential to express answers in this measure, as the exercise requires.

Using radian measure provides a seamless link between the angle and the arc length in a unit circle, simplifying many calculations and making it a preferred measure in higher mathematics.