Problem 10 - Contemporary Precalculus

Question

Find all solutions of the equation. $$\csc x=6.4$$

Step-by-Step Solution

Answer

Question: Solve the equation \(\csc x=6.4\) for all possible values of \(x\). Answer: The solutions for the equation \(\csc x=6.4\) are given by \(x = \arcsin \left(\frac{1}{6.4}\right) + 2\pi n\), where \(n\) is any integer.

Step 1: Convert the equation to sine form

The cosecant function is the reciprocal of the sine function. So, we can write the given equation in terms of the sine function by taking the reciprocal of each side: $$\sin x = \frac{1}{6.4}$$

Step 2: Find the principal solution

To find the principal solution of the equation, we'll use the inverse sine function, which gives us the value of \(x\) in the first rotation of the unit circle (between \(-\pi\) and \(\pi\)): $$x = \arcsin \left(\frac{1}{6.4}\right)$$

Step 3: Find all solutions for x

Since the sine function is periodic with a period of \(2\pi\), we can find all solutions for \(x\) by adding integer multiples of the period to the principal solution: $$x = \arcsin \left(\frac{1}{6.4}\right) + 2\pi n$$ Where \(n\) is any integer. Now, we have found all solutions for the equation: $$\csc x = 6.4$$

Key Concepts

Cosecant Function

The cosecant function, denoted as \(\csc x\), is an important trigonometric function. It's directly related to the sine function, as it is the reciprocal. This means:

• \(\csc x = \frac{1}{\sin x}\)

Cosecant is often used when dealing with right triangles and circles. It's the length of the hypotenuse divided by the length of the opposite side in a right triangle.

Since \(\csc x\) involves division by \(\sin x\), it’s undefined wherever \(\sin x = 0\). This occurs when \(x\) is a multiple of \(\pi\).

When solving equations involving \(\csc x\), you often first convert it to sine to simplify the equation. This involves taking the reciprocal on both sides, making it easier to handle.

Sine Function

The sine function is one of the foundational trigonometric functions. It is defined on the unit circle as the \(y\)-coordinate of a point.

• Here is the basic relationship: \(\sin x = \text{opposite side/hypotenuse}\)

The sine function is periodic, meaning it repeats its values over regular intervals. Specifically, \(\sin x\) has a period of \(2\pi\).

This periodic nature allows us to find infinitely many solutions to equations involving sine by using its periodic formula. For example, if you have \(\sin x = a\), all solutions can be written as \(x = \arcsin(a) + 2\pi n\) where \(n\) is an integer.

Sine is also an odd function, meaning that \(\sin(-x) = -\sin(x)\), a useful property when solving trigonometric equations.

Inverse Sine Function

The inverse sine function, denoted \(\arcsin\), is used to find the angle whose sine is a given number.

• It is the inverse of the sine function within a specific range: \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\).

This range is chosen because it allows for a one-to-one correspondence, which is necessary for defining an inverse.

When solving an equation like \(\sin x = \frac{1}{6.4}\), \(\arcsin\) helps find the principal value, the initial solution within the defined range.

The inverse sine provides a single angle, but since sine is periodic, we often generate additional solutions by adding multiples of the full period \(2\pi\). Thus, all solutions are represented as \(x = \arcsin(a) + 2\pi n\), accommodating the cyclic nature of trigonometric functions.