Problem 74 - Precalculus with Limits

Question

In Exercises 63 - 74, use inverse functions where needed to find all solutions of the equation in the interval \( [0,2\pi) \). \( \csc^2 x - 5 \csc x = 0 \)

Step-by-Step Solution

Answer

The solutions of the given equation in the interval \( [0,2\pi) \) are \( x = \arcsin(1/5) \) and \( x = \pi - \arcsin(1/5) \)

Step 1: Factorize the Quadratic Equation

The given equation is \( \csc^2 x - 5 \csc x = 0 \). Factorize it to \( \csc x (\csc x -5) = 0 \)

Step 2: Solve for \( \csc x \)

In order to find the solutions of \( x \), separately set each factor equal to zero i.e. \( \csc x = 0 \) and \( \csc x = 5 \) and solve for \( x \)

Step 3: Find the Inverse of \( \csc 0 \)

For \( \csc x = 0 \), no solution exists because cosecant function is not defined at \( x = 0 \)

Step 4: Find the Inverse of \( \csc 5 \)

For \( \csc x = 5 \), calculate the arcsin of the reciprocal of 5 to get \( x = \arcsin(1/5) \). But \( \csc x \) function also has a negative period in the second quadrant i.e. \( x = \pi - \arcsin(1/5) \)

Step 5: Find All Solutions within the Interval

The solutions within the interval [0, \( 2\pi \)) are \( x = \arcsin(1/5) \) and \( x = \pi - \arcsin(1/5) \)

Key Concepts

Understanding the Cosecant Function

The cosecant function, denoted as \( \csc x \), is the reciprocal of the sine function. Simply put, \( \csc x = \frac{1}{\sin x} \). The cosecant function is useful when dealing with trigonometric equations, especially when the problem modifies the standard sine relationship. Remember, the cosecant is undefined wherever sine is zero, this occurs at integer multiples of \( \pi \) (i.e., \( 0, \pi, 2\pi \)). This is important because these locations will not provide valid solutions for problems involving the cosecant function.

Regarding graphs, the cosecant function tends to create discontinuous curves as it exhibits vertical asymptotes where the sine function is zero. This can lead to unique solution characteristics in equations that include \( \csc x \). When solving for an equation like \( \csc^2 x - 5 \csc x = 0 \), you need to factor it out to separate the function into manageable parts which can then be solved individually.

Solving Quadratic Equations in Trigonometry

Quadratic equations take the standard form \( ax^2 + bx + c = 0 \). In trigonometry, these can involve trigonometric identities, like \( \csc x \). The given equation \( \csc^2 x - 5 \csc x = 0 \) is a quadratic form disguised in trigonometric clothing. The key to solving it is identifying \( \csc x \) as a factor, allowing you to rewrite it as \( \csc x (\csc x - 5) = 0 \).

To find solutions, set each factor to zero:

• \( \csc x = 0 \): Here, no solutions exist because the cosecant function is undefined for any \( x \) that would make it zero.
• \( \csc x = 5 \): This gives a solvable scenario by using the definition that \( \csc x = \frac{1}{\sin x} \), so \( \sin x = \frac{1}{5} \). You then solve for \( x \) by using inverse functions.

Breaking down complex trigonometric equations into simpler quadratic forms facilitates identifying all possible solutions across a specified interval.

Using the Arcsin Function to Find Angles

The arcsin function, or inverse sine function, is denoted as \( \arcsin(x) \). It helps determine the angle whose sine value corresponds to a given number. Its range is limited to \(-\frac{\pi}{2} \) to \( \frac{\pi}{2} \), to ensure it remains a function (one output for each input).

In our problem \( \csc x = 5 \), which translates to \( \sin x = \frac{1}{5} \), we use \( x = \arcsin(\frac{1}{5}) \) to find the primary angle. Then note that sine has certain symmetrical properties, especially in trigonometric quadrants. For other possible angles in the interval \( [0, 2\pi) \), we consider \( x = \pi - \arcsin(\frac{1}{5}) \).

Utilizing \( \arcsin \) helps us find these corresponding angles systematically and ensures all solutions within the desired range are found. When tackling problems involving arcsin and non-standard intervals, always consider the nature of sine's symmetry and the quadrants where specific sine values are positive or negative.