Problem 19 - Algebra and Trigonometry

Question

Find the exact value of the trigonometric function at the given real number. $$ \begin{array}{lll}{\text { (a) } \csc \left(-\frac{\pi}{2}\right)} & {\text { (b) } \csc \frac{\pi}{2}} & {\text { (c) } \csc \frac{3 \pi}{2}}\end{array} $$

Step-by-Step Solution

Answer

(a) -1, (b) 1, (c) -1

Step 1: Understanding Cosecant

The cosecant function, \(\csc(\theta)\), is the reciprocal of the sine function, i.e., \(\csc(\theta) = \frac{1}{\sin(\theta)}\). To find \(\csc\) of any angle, we first need to determine the sine of that angle.

Step 2: Calculate \(\csc(-\frac{\pi}{2})\)

First, find \(\sin(-\frac{\pi}{2})\). Using the sine function, we know that \(\sin(-\frac{\pi}{2}) = -1\). Therefore, \(\csc(-\frac{\pi}{2}) = \frac{1}{-1} = -1\).

Step 3: Calculate \(\csc(\frac{\pi}{2})\)

Next, find \(\sin(\frac{\pi}{2})\). From trigonometric identities, \(\sin(\frac{\pi}{2}) = 1\). Thus, \(\csc(\frac{\pi}{2}) = \frac{1}{1} = 1\).

Step 4: Calculate \(\csc(\frac{3\pi}{2})\)

Finally, determine \(\sin(\frac{3\pi}{2})\). We know that \(\sin(\frac{3\pi}{2}) = -1\). Therefore, \(\csc(\frac{3\pi}{2}) = \frac{1}{-1} = -1\).

Key Concepts

Cosecant Function

The cosecant function is a lesser-known trigonometric function that often plays an essential role in solving various mathematical problems. Represented as \( \csc(\theta) \), it is the reciprocal of the sine function. This simply means that to determine the value of the cosecant, you take the inverse of the sine of an angle: \[ \csc(\theta) = \frac{1}{\sin(\theta)} \] This relationship highlights why understanding the sine function is imperative when working with the cosecant. If the sine value is zero, the cosecant is undefined, as division by zero is not possible. This function is often applied in scenarios where we need to consider vertical directions or determine certain angular relationships.
Keep in mind that the behavior of the cosecant mirrors the sine function due to its reciprocal nature.

Sine Function

The sine function is a foundational concept in trigonometry, often introduced early in mathematics education. Represented as \( \sin(\theta) \), it describes the y-coordinate of a point on the unit circle—this is crucial because it allows us to translate angles into concrete numerical values. For example:

• \( \sin\left(\frac{\pi}{2}\right) = 1 \)
• \( \sin(0) = 0 \)
• \( \sin\left(-\frac{\pi}{2}\right) = -1 \)

These values are fundamental in determining the corresponding cosecant values.
Remember that the sine function oscillates between -1 and 1, mapping every real number to this range. This sinusoidal wave is pivotal in various fields, including physics, engineering, and even music theory, serving as a model for periodic phenomena.

Reciprocal Trigonometric Identities

Reciprocal trigonometric identities are relationships that express trigonometric functions as the reciprocals of each other. They form vital connections between different trigonometric functions. In addition to the cosecant-sine relationship, here are a few other common identities:

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)
• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)

These identities are particularly useful in simplifying trigonometric expressions or solving equations.
Not only do they reduce computational complexity, but they also deepen understanding of how trigonometric functions interact as different angles change in magnitude. Understanding these identities allows students to tackle more advanced problems with clarity and confidence.

Angle Measurement in Radians

Angle measurement can be expressed in degrees or radians, with radians being the standard in most mathematical contexts. Radians measure angles based on the radius of the circle. This unit ties directly to the unit circle's geometry, where one full rotation around the circle equals \(2\pi\) radians. This unit of measure is essential in understanding the periodicity and the function value calculations of trigonometric functions. For instance, in solving problems for \( \csc\left(\theta\right) \), it’s vital to grasp that \(-\frac{\pi}{2}, \frac{\pi}{2},\) and \(\frac{3\pi}{2}\) correspond to specific points on the unit circle, affecting their sine values accordingly.
By familiarizing yourself with radians, you'll appreciate the seamless nature they offer in linking trigonometry and calculus. This is especially critical when dealing with derivative and integral calculations involving trigonometric functions.