Problem 7 - Precalculus with Calculus Previews

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Function

The problem requires us to find the value of the cosecant function, \( \csc \theta = \frac{1}{\sin \theta} \), for an angle of \( -\frac{\pi}{3} \). Cosecant is the reciprocal of the sine function.

Step 2: Determine the Reference Angle

Since we are dealing with a negative angle, \( \theta = -\frac{\pi}{3} \), we first find its reference angle. The reference angle for \( -\frac{\pi}{3} \) is \( \frac{\pi}{3} \), because angles in the first and fourth quadrants have reference angles that are positive.

Step 3: Find the Sine of the Reference Angle

The sine of \( \frac{\pi}{3} \) is \( \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \). Since \( -\frac{\pi}{3} \) is in the fourth quadrant, where sine is negative, \( \sin \left( -\frac{\pi}{3} \right) = -\frac{\sqrt{3}}{2} \).

Step 4: Calculate the Cosecant

Using the sine value obtained, find the cosecant: \( \csc \left( -\frac{\pi}{3} \right) = \frac{1}{\sin \left( -\frac{\pi}{3} \right)} = \frac{1}{-\frac{\sqrt{3}}{2}} = -\frac{2}{\sqrt{3}} \).

Step 5: Simplify the Result

To rationalize the denominator, multiply the numerator and the denominator by \( \sqrt{3} \): \( -\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = -\frac{2\sqrt{3}}{3} \). Hence, \( \csc \left(-\frac{\pi}{3}\right) = -\frac{2\sqrt{3}}{3} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in understanding angles and their relationships in mathematics. They form the cornerstone of trigonometry. Among these functions, the cosecant (\( \csc \theta \) is significant. The cosecant function is essentially the reciprocal of the sine function, given by:

• \[ \csc \theta = \frac{1}{\sin \theta} \]

This means that to find the value of the cosecant for any angle \( \theta \), you just take the reciprocal of the sine value of that angle. This makes the cosecant function sensitive to the sine value being zero because the reciprocal of zero is undefined.
When working with trig functions like this, it's helpful to understand their behavior in different quadrants of the unit circle. Knowing the sine value of common angles, such as \( \frac{\pi}{3} \) or \( \frac{\pi}{6} \), also aids immensely in manual calculations. For example, for \( \frac{\pi}{3} \), the sine value is \( \frac{\sqrt{3}}{2} \), making its cosecant \( \frac{2}{\sqrt{3}} \). Understanding these relationships allows for quick calculations and can even help you solve problems without a calculator.

Rationalizing Denominators

Rationalizing the denominator is a crucial process in simplifying expressions in mathematics. It involves altering a fraction so that there are no radicals in the denominator. For instance, if the fraction is \( \frac{1}{\sqrt{3}} \), the operation would be to multiply both the numerator and the denominator by \( \sqrt{3} \) resulting in:

• \[ \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \]

This step is crucial, especially in trigonometric calculations where leaving radicals in the denominator is not ideal. This method simplifies the result to a more acceptable form for further mathematical operations.
In our specific exercise, computing \( \csc \left( -\frac{\pi}{3} \right) \), we initially find \(-\frac{2}{\sqrt{3}} \). By multiplying with \( \frac{\sqrt{3}}{\sqrt{3}} \), it becomes \(-\frac{2\sqrt{3}}{3} \). This rationalization not only simplifies the expression but also maintains the accuracy of the computation. Although it might appear as a small step, rationalizing denominators is a valuable technique that makes complex calculations more manageable.

Reference Angles

Reference angles are essential when dealing with trigonometric functions involving negative angles or those greater than \( 2\pi \). A reference angle is formed by the angle and the horizontal axis. It helps us understand angles in any quadrant using positive acute angles.
To find a reference angle, consider:

• Positive angles: Take the angle modulo \( 2\pi \) if it's greater than \( 2\pi \).
• Negative angles: Convert the angle to positive by adding \( 2\pi \) until positive.

In this exercise, \( -\frac{\pi}{3} \) lies in the fourth quadrant. Its reference angle is \( \frac{\pi}{3} \) since it directly corresponds to the smallest angle made with the x-axis.
Using reference angles simplifies determining trigonometric function values. The cosine and sine of any angle in the fourth quadrant will have the same magnitude as those for its reference angle in the first quadrant, but differentiated by the sign based on quadrant rules. For sine, values are negative in the fourth quadrant, which was crucial in our solution to find \( \sin(-\frac{\pi}{3}) = -\frac{\sqrt{3}}{2} \), enabling the calculation of the corresponding cosecant value.