Problem 42 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\(\theta = \pi\) degrees.

Step 1: Understanding the Problem

First, we need to find the angle \( \theta \) such that the cosecant of \( \theta \) is \(-1\). This means \( \csc \theta = -1 \). We know that \( \csc \theta = \frac{1}{\sin \theta} \), so we need \( \sin \theta = -1 \).

Step 2: Identify Possible Angles

The sine of an angle equals \(-1\) at specific angles. In radians, this occurs at an odd multiple of \(\frac{3\pi}{2}\): \( \theta = \frac{3\pi}{2} + 2k\pi \), where \( k \) is any integer. We want to find the degree measure of such an angle.

Step 3: Find the Degree Measure

We will convert the radian measure \( \frac{3\pi}{2} \) to degrees. We know that \( \pi \) radians is equivalent to \( 180^\circ \). So, \( \frac{3\pi}{2} = \frac{3 \times 180}{2} = 270^\circ \).

Step 4: Check the Interval for Inverse Cosecant

The principal value for \( \csc^{-1}(x) \) is between \(-\frac{\pi}{2}, \frac{\pi}{2}\) not including 0, (for positive \( x\)) or between \( 0, \pi \) (for negative \( x\)). Since \( x = -1 \), we choose \( 270^\circ - 360^\circ \) which aligns with the range given, becoming \( -90^\circ \) or equivalently \( \theta = \pi \).

Key Concepts

Cosecant Function

The Cosecant Function, denoted as \ \( \csc \theta \ \), is one of the six fundamental trigonometric functions. It is closely related to the sine function as it is defined as the reciprocal of sine. This relationship to sine can be expressed in the equation:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Understanding this function helps in solving problems involving inverse trigonometric functions, just like the one in the exercise we discussed.
When you're given an inverse cosecant, \ \( \csc^{-1}(x) \ \), your task is to find the angle whose cosecant is \ \( x \ \).
This requires thinking about sine, because when you know that \ \( \csc \theta = x \ \), it tells us that \ \( \sin \theta = \frac{1}{x} \ \).
The tricky part is understanding the domains and ranges for these functions. The fundamental range for \ \( \csc^{-1}(x) \ \) is generally between two intervals:

• \( -\frac{\pi}{2} < \theta < \frac{\pi}{2} \) for positive \ \( x \ \)
• \( 0 < \theta < \pi \) for negative \ \( x \ \) (not including 0)

Angle Measurement

Understanding angle measurements is crucial when dealing with trigonometry and inverse trigonometric functions. Angles can be measured in two main units: degrees and radians.
Degrees are a more familiar unit for most people, as a full circle is 360 degrees. In trigonometry, the quadrant of an angle significantly influences the properties of sine, cosine, and other functions.
Radians, on the other hand, are the standard unit in mathematical calculations involving trigonometry because they directly relate to the arc length of a circle. The radian measure of an angle is defined by the formula:

• \( \text{Radians} = \frac{\text{Arc Length}}{\text{Radius}} \)

To convert between these two, remember that \( \pi \) radians equal 180 degrees. Hence, one radian is approximately 57.2958 degrees.
This conversion helps when you need to solve an exercise requiring angles to be in a different unit, like converting \( \frac{3\pi}{2} \) radians into degrees, resulting in 270 degrees.
This understanding assisted us in interpreting the exercise where \( \theta \) had to be in degrees for easier comprehension or specific problem requirements.

Radians and Degrees

It's common in mathematics to swap between radians and degrees since different problems demand one type of measurement over the other. Understanding the connection between these two systems can significantly aid your problem-solving skills in trigonometry.
The degree measure provides a segmented view of circles and angles, ranging from 0 to 360 degrees. This is often more intuitive when considering navigation or sectors of circular graphs.
On the contrary, radians give you a measurement based on the radius of the circle. Therefore:

• A full circle equals \( 2\pi \) radians.
• A semicircle equals \( \pi \) radians.
• A quarter-circle or "right angle" is \( \frac{\pi}{2} \) radians.

Remember these conversions:

• \( 180^{\circ} = \pi \) radians
• \( 90^{\circ} = \frac{\pi}{2} \) radians
• \( 270^{\circ} = \frac{3\pi}{2} \) radians

These conversions are pivotal when an exercise or solution needs an angle to be in a particular format or when interpreting the principal values of inverse trigonometric functions, just like the discussed problem where we identified the \( \csc^{-1}(x) \) value in degrees.