Problem 7 - Precalculus

Question

In Exercises 1 - 20 , find the exact value or state that it is undefined. $$ \csc \left(-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

The exact value of \( \csc(-\frac{\pi}{3}) \) is \( -\frac{2\sqrt{3}}{3} \).

Step 1: Understand the Function

The function given is \( \csc(-\frac{\pi}{3}) \). The cosecant function, \( \csc(x) \), is the reciprocal of the sine function: \( \csc(x) = \frac{1}{\sin(x)} \). To find \( \csc(-\frac{\pi}{3}) \), we need to find \( \sin(-\frac{\pi}{3}) \) first.

Step 2: Determine the Sine Value

Use the property of the sine function that \( \sin(-x) = -\sin(x) \). For \( \sin(-\frac{\pi}{3}) \), we have:\[ \sin(-\frac{\pi}{3}) = -\sin(\frac{\pi}{3}). \] We know that \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \), so \( \sin(-\frac{\pi}{3}) = -\frac{\sqrt{3}}{2}. \)

Step 3: Calculate the Cosecant Value

Now find the cosecant by taking the reciprocal of the sine value: \( \csc(-\frac{\pi}{3}) = \frac{1}{\sin(-\frac{\pi}{3})} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} \).

Step 4: Simplify the Expression

To simplify \( -\frac{2}{\sqrt{3}} \), rationalize the denominator: \[ -\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3}. \] Thus, the exact value is \( -\frac{2\sqrt{3}}{3} \).

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc(x) \), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the sine function. Simply put, \( \csc(x) = \frac{1}{\sin(x)} \). This means that the cosecant gives us the ratio of the hypotenuse to the opposite side in a right triangle.
The cosecant function is especially useful in scenarios where the hypotenuse and opposite side are known, but the angle is needed. Unlike the sine function, the cosecant function is undefined whenever sine equals zero. This occurs at specific angles such as \( 0 \), \( \pi \), \( 2\pi \), etc.
In terms of its graph and behavior, \( \csc(x) \) is periodic with a period of \( 2\pi \). It has vertical asymptotes at points where \( \sin(x) = 0 \). This results in gaps at those points in its graph.

Sine Function

The sine function, denoted as \( \sin(x) \), is a fundamental concept in trigonometry. It represents the ratio of the length of the opposite side to the hypotenuse in a right triangle.
Mathematically, the value of \( \sin(x) \) ranges from -1 to 1. It is periodic, with a period of \( 2\pi \), indicating that the pattern of values repeats every \( 2\pi \) radians. This repetitive nature makes the sine function a wave-like, continuous curve.

• For example, \( \sin(\pi/3) = \frac{\sqrt{3}}{2} \).
• Additionally, an important property to note is that \( \sin(-x) = -\sin(x) \), showcasing the symmetry of the sine function about the origin.

Understanding how the sine function behaves is crucial, as it is extensively used in various applications like calculating sound waves, alternating current circuits, and even in modeling tides in the ocean.

Reciprocal Identities

Reciprocal identities link one trigonometric function to another by showing the connection between a function and its reciprocal. Particularly, the cosecant function is tied to the sine function through its reciprocal identity:
\( \csc(x) = \frac{1}{\sin(x)} \).
Understanding reciprocal identities allows for easier simplification of expressions and solving equations. Since the sine of an angle is often easier to compute and remember, using its reciprocal can simplify computations.

• For instance, when asked to find \( \csc(x) \), you may first compute \( \sin(x) \) and then take its reciprocal.
• Memorizing these identities also assists with quickly accessing values without explicitly solving for each function individually.

In mathematical practice, these identities enable seamless transitions between different trigonometric functions and play a vital role in trigonometry.

Angle Properties

When dealing with trigonometric functions, understanding angle properties is essential. Angles can be positive or negative, and they can also extend beyond the traditional 0 to \( 2\pi \) radian range.
A vital property of angles is that the trigonometric functions can handle negative angles as well. For the sine function, this is exhibited by the property \( \sin(-x) = -\sin(x) \). This reflection across the x-axis shows the symmetry of trigonometric functions.

• For instance, when calculating \( \sin(-\frac{\pi}{3}) \), we use the property to find that it equals \(-\sin(\frac{\pi}{3}) \).
• Another essential aspect of angles is their unit. When stated in radians, they can effortlessly translate between angular and linear measurements.

The ability to work with angles in different quadrants and recognizing their cyclic properties enables us to solve more complex trigonometric problems efficiently.