Problem 4 - Precalculus with Calculus Previews

Question

Find the indicated value without the use of a calculator. $$ \csc \left(-\frac{3 \pi}{2}\right) $$

Step-by-Step Solution

Answer

\(\csc\left(-\frac{3 \pi}{2}\right) = 1\).

Step 1: Understand the Reciprocal Identity

The cosecant function is the reciprocal of the sine function. Therefore, \( \csc(\theta) = \frac{1}{\sin(\theta)} \). To find \( \csc \left(-\frac{3 \pi}{2}\right) \), we need to first find \( \sin \left(-\frac{3 \pi}{2}\right) \).

Step 2: Determine the Reference Angle

\(-\frac{3\pi}{2}\) is a multiple of \( \pi \), specifically \(-\frac{3\pi}{2} = -270^\circ\). Since \(-270^\circ\) is counterclockwise equivalent to \(90^\circ\) (because \(-270^\circ = 360^\circ - 270^\circ\)), the reference angle is \(90^\circ\) or \(\frac{\pi}{2}\) radians.

Step 3: Evaluate the Sine Function

The sine of \(-\frac{3\pi}{2}\), which is the same as \(\sin(90^\circ)\), can be found by using the reference angle. \(\sin(90^\circ) = 1\), and hence \( \sin \left(-\frac{3 \pi}{2}\right) = 1 \) since sine is odd and preserved when shifted by full rotations.

Step 4: Calculate the Cosecant

Using the reciprocal identity \( \csc \left(-\frac{3 \pi}{2}\right) = \frac{1}{\sin \left(-\frac{3 \pi}{2}\right)} \), and knowing that \( \sin \left(-\frac{3 \pi}{2}\right) = 1 \), it follows that \( \csc \left(-\frac{3 \pi}{2}\right) = \frac{1}{1} = 1 \).

Key Concepts

Reciprocal Identity

The concept of reciprocal identities is essential in trigonometry, and it simplifies the process of finding functions like cosecant, secant, and cotangent. When we talk about reciprocal identities, we focus on how one trigonometric function is the inverse, or reciprocal, of another. For cosecant, the reciprocal identity is established as follows:

• The cosecant function, denoted as \( \csc(\theta) \), is defined as the reciprocal of the sine function.
• This means that \( \csc(\theta) = \frac{1}{\sin(\theta)} \).

In the original problem, understanding the reciprocal identity is the first step to finding \( \csc \left(-\frac{3 \pi}{2}\right) \). Initially, we must determine the sine of the angle \( -\frac{3 \pi}{2} \). Only then can we apply the reciprocal identity to find the desired cosecant value.Utilizing these reciprocal relationships not only aids in calculations but also in grasping the broader symmetry and periodicity of trigonometric functions.

Reference Angle

The reference angle is a crucial concept in trigonometry that helps us find the sine, cosine, and tangent of complex angles using known values from the first quadrant of the unit circle. A reference angle is the smallest angle that the terminal side of a given angle makes with the x-axis.Here’s how you can determine reference angles:

• Convert angles into a positive equivalent if needed by adding or subtracting full rotations (360° or \(2\pi\)).
• The reference angle is always positive and less than \(90^\circ\) or \(\frac{\pi}{2}\).

In the problem, \(-\frac{3\pi}{2}\) is exactly \(-270^\circ\), which corresponds to the angle closest to \(90^\circ\) in the first quadrant after a full rotation. By recognizing this, we understand that the trigonometric functions for \(-270^\circ\) and \(90^\circ\) are equivalent, guiding us to find the sine value effectively for the calculation.

Sine Function

The sine function, one of the primary trigonometric functions, is defined on the unit circle. It measures the vertical coordinate of a point moving around the circle. For any angle \(\theta\), \(\sin(\theta)\) represents the y-coordinate of the point where the terminal side of the angle intersects the unit circle.Key features of the sine function:

• It is periodic with a period of \(2\pi\), meaning \( \sin(\theta + 2\pi) = \sin(\theta) \).
• The sine function is odd, which implies that \(\sin(-\theta) = -\sin(\theta)\).

In our example, the angle \(-\frac{3\pi}{2}\) (or \(-270^\circ\)) can be represented as multiple full rotations from the reference \(90^\circ\), keeping the sine value unchanged due to periodicity.When calculated, \(\sin(90^\circ) = 1\). Since the sine function value has remained consistent through rotations and shifts, \(\sin(-\frac{3\pi}{2}) = 1\). This outcome is crucial for determining the cosecant value using the reciprocal identity.