Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Find the exact value of each real number \(y .\) Do not use a calculator. $$y=\csc ^{-1} 2$$

Step-by-Step Solution

Answer

\( y = \frac{\pi}{6} \)

Step 1: Understand the Problem

We need to find the real number \( y \) such that \( \csc y = 2 \). This involves understanding the relationship between the cosecant and sine functions.

Step 2: Relationship between Cosecant and Sine

Recall that the cosecant function is the reciprocal of the sine function: \( \csc y = \frac{1}{\sin y} \). Therefore, if \( \csc y = 2 \), then \( \frac{1}{\sin y} = 2 \).

Step 3: Solve for \( \sin y \)

From the equation \( \frac{1}{\sin y} = 2 \), we can multiply both sides by \( \sin y \) and divide by 2 to get: \( \sin y = \frac{1}{2} \).

Step 4: Identify the Angle \( y \)

The exact values for \( y \) occur at angles where \( \sin y = \frac{1}{2} \). These are common angles on the unit circle. \( y = \frac{\pi}{6} \) or \( 5\frac{\pi}{6} \) are the angles where \( \sin y = \frac{1}{2} \) and are within the range of the \( \csc^{-1} \) function, which is \( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \). However, since \( \csc \) is undefined at \( y = 0 \), only \( \frac{\pi}{6} \) is valid.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in trigonometry. They allow us to work backwards from the value of a trigonometric function to find the angle that would produce that function value.

• The family of inverse trigonometric functions includes arcsin, arccos, arctan, as well as the less commonly used arcsec, arccsc, and arccot.
• These functions designate the angle from a given ratio, such as a sine or cosine value.

When dealing specifically with the inverse cosecant function, denoted as \( ext{csc}^{-1} \), we reverse the operation of the cosecant. Rather than finding the reciprocal of a sine value as in the cosecant, \( ext{csc}^{-1} \) asks: "What angle gives me this reciprocal value when applied to the sine function?"
An important thing to note is the restricted range of these functions. For example, \( ext{csc}^{-1} \) is typically defined over the domain \( (-\infty, -1] \cup [1, \infty) \), and its range is \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), excluding values that would make the function undefined, such as 0.

Cosecant Function

The cosecant function, denoted as \( ext{csc} \), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function, emphasizing its unique role in trigonometry.

• Mathematically, this is represented as \( ext{csc}(y) = \frac{1}{\sin(y)} \).
• This means whenever \( ext{csc} \) is involved, it requires that the sine value not be zero, as division by zero is undefined.

In our problem, we're given \( ext{csc}^{-1}(2) \), which converts to finding an angle \( y \) such that \( ext{csc}(y) = 2 \) or \( \frac{1}{\sin(y)} = 2 \). This implies finding when \( \sin(y) = \frac{1}{2} \). As such, understanding cosecant means grasping its reciprocal nature, which simplifies finding solutions on the unit circle.

Unit Circle

The unit circle is an incredibly valuable tool for understanding trigonometric functions and their values across different angles. It is a circle with a radius of 1 centered at the origin of a coordinate plane.

• It helps visualizing the relationship between angles and trigonometric ratios like sine, cosine, and tangent.
• On the unit circle, the sine of an angle is the y-coordinate of the corresponding point, and the cosine is the x-coordinate.
• Knowing specific angles and their trigonometric values on the unit circle is crucial for solving inverse trigonometric problems.

In our case, when evaluating \( \csc^{-1}(2) \), we identified that \( \sin y = \frac{1}{2} \).
On the unit circle, this occurs at \( y = \frac{\pi}{6} \) or \( y = 5\frac{\pi}{6} \). However, within the typical range of \( \csc^{-1} \), which spans \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \) without hitting precisely at zero, \( y = \frac{\pi}{6} \) is the valid output.