Problem 73 - Precalculus Mathematics for Calculus

Question

Find all solutions of the equation that lie in the interval \([0, \pi] .\) State each answer correct to two decimal places. $$\csc x=3$$

Step-by-Step Solution

Answer

The solutions are x ≈ 0.34 and x ≈ 2.80 radians.

Step 1: Understand the Relationship

Recall that the cosecant function is the reciprocal of the sine function. Therefore, if \( \csc x = 3 \), it follows that \( \sin x = \frac{1}{3} \).

Step 2: Determine the Interval

We are asked to find solutions in the interval \([0, \pi]\). This interval includes the domain for which the sine function is positive, between 0 and \( \pi \).

Step 3: Use Inverse Sine Function

To solve for \( x \), use the inverse sine function: \( x = \arcsin\left(\frac{1}{3}\right) \). Compute this using a calculator to ensure accuracy.

Step 4: Calculate the Principal Solution

Calculate \( x = \arcsin\left(\frac{1}{3}\right) \). Enter \( \frac{1}{3} \) into the calculator and find the angle whose sine is \( \frac{1}{3} \). This gives \( x \approx 0.34 \) radians.

Step 5: Consider Symmetry in the Interval

Since the interval is \([0, \pi]\), check for other possible solutions. The sine function is symmetric about \( \pi/2 \). Therefore, another solution can be found as \( x = \pi - \arcsin\left(\frac{1}{3}\right) \).

Step 6: Calculate the Symmetric Solution

Compute the symmetric solution: \( x = \pi - 0.34 \), which yields \( x \approx 2.80 \) radians.

Step 7: Verify Solutions

Ensure that both solutions \( x \approx 0.34 \) and \( x \approx 2.80 \) are within the interval \([0, \pi]\) and have \( \csc x = 3 \).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc x \), is an important concept in trigonometry. It is defined as the reciprocal of the sine function, which means that \( \csc x = \frac{1}{\sin x} \). Because of this reciprocal relationship, the cosecant function is undefined whenever the sine function is zero. Understanding the cosecant function is crucial when solving trigonometric equations. It often requires converting a cosecant equation to a sine equation, as demonstrated in the exercise where we start with \( \csc x = 3 \) and rewrite it as \( \sin x = \frac{1}{3} \). This step is vital for finding the solutions within a specified interval.

Sine Function

The sine function, represented as \( \sin x \), is one of the foundational trigonometric functions. It describes the relationship between an angle in a right triangle and the ratio of the length of the opposite side to the hypotenuse.

• Sine function values range between -1 and 1.
• The sine wave starts at zero, rises to its maximum at \( \pi/2 \), returns to zero by \( \pi \), and is symmetric over an interval from \( 0 \) to \( 2\pi \).

In this context, the sine function helps to determine angles that satisfy particular conditions, like when we solved \( \sin x = \frac{1}{3} \) using the inverse sine function. Sine's periodic nature also allows us to explore solutions that might not be immediately apparent but within the given interval.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential in finding angles from given trigonometric values. The inverse sine function, denoted as \( \arcsin \) or \( \sin^{-1} \), provides the angle whose sine is a specific number.

• \( \arcsin \) returns a value between \(-\pi/2\) and \(\pi/2\).
• In our exercise, we calculated \( x = \arcsin\left(\frac{1}{3}\right) \).
• This angle-specific approach becomes critical when dealing with trigonometric equations, helping convert a problem into a calculable format.

When solving \( \sin x = \frac{1}{3} \), using \( \arcsin \) gave us the principal solution, which we found using a calculator to an approximate degree of accuracy, \( x \approx 0.34 \) radians. This methodological aspect of identifying solutions within a range hinges on the inverse function's concept.

Symmetry in Trigonometric Functions

Symmetry plays a significant role in trigonometric functions, aiding in finding all potential solutions in specified intervals. The sine function is symmetric about the line \( x = \pi/2 \), and this characteristic helps identify additional solutions that aren't immediately obvious from straightforward calculations.

• In the interval \([0, \pi]\), the symmetry property of sine suggests complementary angles that also satisfy the equation.
• For a given angle \( x \), another potential solution could be \( \pi - x \).
• This principle was employed to discover the second solution, \( x \approx 2.80 \) radians, from the symmetric equation \( \pi - \arcsin\left(\frac{1}{3}\right) \).

Understanding the symmetrical nature of trigonometric functions permits the identification of alternate solutions within specified constraints, encompassing the full range of possibilities inherent to the function's behavior.