Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Give the degree measure of \(\theta,\) if it exists. Do not use a calculator. $$\theta=\csc ^{-1}(-2)$$

Step-by-Step Solution

Answer

\( \theta = -30^\circ \).

Step 1: Understanding the Problem

We are asked to find the angle \( \theta \) in degrees given the expression \( \theta = \csc^{-1}(-2) \). \( \csc^{-1}(x) \) is the inverse cosecant function, which means it finds an angle \( \theta \) such that \( \csc(\theta) = x \). Here, we need \( \csc(\theta) = -2 \).

Step 2: Recall the Cosecant Definition

The cosecant function is defined as \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Therefore, if \( \csc(\theta) = -2 \), then \( \sin(\theta) = -\frac{1}{2} \).

Step 3: Determine Possible Angles for \( \sin(\theta) = -\frac{1}{2} \)

The sine of an angle equals \(-\frac{1}{2}\) at specific angles. In radians, these angles are \( \theta = \frac{7\pi}{6} \) and \( \theta = \frac{11\pi}{6} \) within the principal range of the cosecant function, which is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

Step 4: Convert Radian to Degrees

To find the corresponding degree measures, we convert \( \theta = \frac{7\pi}{6} \) and \( \theta = \frac{11\pi}{6} \) from radians to degrees by multiplying with \( \frac{180}{\pi} \):- \( \frac{7\pi}{6} \) converts to \( 210^\circ \).- \( \frac{11\pi}{6} \) converts to \( 330^\circ \).Of these, \( 330^\circ \) corresponds to the smallest negative range conventionally used from the principal value \([-90^\circ, 90^\circ]\) in inverse functions.

Step 5: Choose the Principal Value

The principal value for \( \csc^{-1} \) determines the angle in \([-90^\circ, 90^\circ]\) where the angle \( \theta \) gives the required value \( \csc(\theta) = -2 \). The valid angle complying with this range is \( -30^\circ \), representing \( \theta = \frac{11\pi}{6} \), because it's equivalent to moving to the negative side from 0 just like \( 330^\circ \) which falls in the fourth quadrant.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc(\theta) \), plays a crucial role in trigonometry as one of the basic trigonometric functions. It is the reciprocal of the sine function, meaning that for a given angle \( \theta \), the cosecant is defined as \( \csc(\theta) = \frac{1}{\sin(\theta)} \). This implies that when \( \sin(\theta) \) is zero, \( \csc(\theta) \) will be undefined since division by zero is not possible.

• The cosecant function is periodic, with a period of \( 2\pi \) radians or \( 360^\circ \).
• It is primarily used in situations where we need to evaluate expressions involving the reciprocal of the sine value.
• Keep in mind that since \( \csc(\theta) \) is the reciprocal, if \( \csc(\theta) = x \), then \( \sin(\theta) = \frac{1}{x} \), unless \( x = 0 \).

Angle Conversion

Trigonometry often requires switching between radians and degrees, which are two different measures for angles. Both radians and degrees are used to measure angles, but choosing the correct unit depends on the context of a given problem.

The conversion between radians and degrees follows an essential relationship, whereby \( 1\) radian equals \( \frac{180}{\pi} \) degrees. Therefore, to convert an angle from radians to degrees, you multiply the radian measure by \( \frac{180}{\pi} \). Conversely, to go from degrees to radians, multiply by \( \frac{\pi}{180} \). This conversion is fundamental in ensuring angles are interpreted correctly in both theoretical and practical aspects of trigonometry.

• Knowing which unit your solution requires can often be guided by the problem's context.
• While one full rotation is \( 360^\circ \), in radians, it amounts to \( 2\pi \).
• For example, if you have an angle such as \( \frac{7\pi}{6} \) and need to convert it to degrees, calculate \( \frac{7\pi}{6} \times \frac{180}{\pi} = 210^\circ \).

Principal Value

Understanding the concept of the principal value is integral when dealing with inverse trigonometric functions. The principal value provides a way to uniquely specify angles obtained from inverse trigonometric functions within a specific range.

For the inverse cosecant function, \( \csc^{-1} \), this principal range is usually chosen between \([-90^\circ, 90^\circ]\) or \([-\frac{\pi}{2}, \frac{\pi}{2}]\). This ensures each output corresponds to one distinct and conventionally accepted angle.

• The principal value helps resolve ambiguities, providing a clear indication of which quadrant the angle resides.
• In scenarios where \( \csc(\theta) = -2 \), the potential angles of \( \frac{7\pi}{6} \) or \( \frac{11\pi}{6} \) must conform to the principal range.
• Assigning \( -30^\circ \) for \( \csc^{-1}(-2) \) is done because it falls within the principal value range and accurately describes a position in the fourth quadrant starting from zero.