Problem 16 - Algebra 2 and Trigonometry

Question

For each function value, write the value or tell why it is undefined. Do not use a calculator. \(\csc \left(-\frac{9 \pi}{2}\right)\)

Step-by-Step Solution

Answer

The value of \(\csc\left(-\frac{9\pi}{2}\right)\) is \(-1\).

Step 1: Understand the Function

The function given is the cosecant function, represented by \(\csc(\theta)\). Recall that \(\csc(\theta) = \frac{1}{\sin(\theta)}\) for any angle \(\theta\). Thus, \(\csc(\theta)\) is undefined where \(\sin(\theta) = 0\).

Step 2: Simplify the Angle

The angle given is \(-\frac{9\pi}{2}\). Since sine and cosine functions are periodic with a period of \(2\pi\), we can add or subtract \(2\pi\) until the angle falls within the range \([0, 2\pi)\). We have:\[-\frac{9\pi}{2} + 4\pi = -\frac{\pi}{2}\] Adding another \(2\pi\):\[-\frac{\pi}{2} + 2\pi = \frac{3\pi}{2}\] So, \(-\frac{9\pi}{2}\) is coterminal with \(\frac{3\pi}{2}\).

Step 3: Evaluate Sine at the Simplified Angle

Now we evaluate \(\sin\left(\frac{3\pi}{2}\right)\). The angle \(\frac{3\pi}{2}\) is directly downward on the unit circle, corresponding to a sine value of \(-1\).

Step 4: Calculate Cosecant

Since \(\csc(\theta) = \frac{1}{\sin(\theta)}\) and \(\sin\left(\frac{3\pi}{2}\right) = -1\), we have:\[\csc\left(\frac{3\pi}{2}\right) = \frac{1}{-1} = -1\] Thus, \(\csc\left(\frac{3\pi}{2}\right) = -1\), so \(\csc\left(-\frac{9\pi}{2}\right) = -1\).

Key Concepts

Trigonometric Identities

Trigonometric identities are foundational tools in trigonometry that allow us to relate various trigonometric functions, such as sine, cosine, and cosecant. The **cosecant function** is closely tied to the sine function through its identity:- **Cosecant as the Reciprocal of Sine**: \(\csc(\theta) = \frac{1}{\sin(\theta)}\) This means the cosecant of an angle \(\theta\), is undefined when the sine of that angle is zero.Understanding these reciprocal identities helps us determine whether trigonometric functions are defined or not at various angles. If we know \(\sin(\theta) = 0\), then \(\csc(\theta)\) cannot exist because division by zero is undefined. Such identities simplify the process of finding values for trigonometric functions without relying on calculators.

Unit Circle

The unit circle is a diagram that represents all the possible angles and their corresponding trigonometric values in a circle with a radius of one unit. This circle is not only a key learning tool but also a powerful concept in solving trigonometric problems by visualization. The coordinates on the unit circle correspond to the cosine and sine values of an angle:- **Horizontal Axis (Cosine):** The x-coordinate on the unit circle.- **Vertical Axis (Sine):** The y-coordinate on the unit circle.The angle \(\frac{3\pi}{2}\), used in the exercise, points directly downward on the unit circle, at the position \((0, -1)\). The sine of \(\frac{3\pi}{2}\) is the y-coordinate, which is \(-1\). When analyzing cosecant, knowing that this angle's sine is \(-1\), we can use the reciprocal identity to quickly find that \(\csc\left(\frac{3\pi}{2}\right) = -1\). The unit circle can also help us remember or find sine and cosine values at multiple angles.

Angle Simplification

Simplifying angles is important due to the periodic nature of trigonometric functions, where they repeat after a certain interval. This is especially true for sine and cosine, which have a period of \(2\pi\):- **Periodic Nature**: Adding or subtracting \(2\pi\) from any angle moves it around the circle but retains equivalent trigonometric values.In the solution of our exercise, we started with the angle \(-\frac{9\pi}{2}\). This angle was simplified using the periodic property:- We added \(4\pi\) to land on an equivalent angle within the principal range, resulting in \(-\frac{\pi}{2}\).- Adding another \(2\pi\) led us to \(\frac{3\pi}{2}\).This coterminal angle simplification allows us to evaluate trigonometric functions at angles within an understandable range, simplifying calculations and understanding.