Problem 194 - Precalculus

Question

For the following exercises, find the exact value of each expression. $$ \csc \frac{\pi}{3} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the function

The cosecant function, \( \csc(\theta) \), is the reciprocal of the sine function. This means that if we know \( \sin(\theta) \), we can find \( \csc(\theta) \) by taking \( \frac{1}{\sin(\theta)} \).

Step 2: Identify the angle

The problem asks for \( \csc \frac{\pi}{3} \). The angle \( \frac{\pi}{3} \) radians is equivalent to 60 degrees.

Step 3: Recall the sine value for the angle

The sine of \( \frac{\pi}{3} \) is \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \). This is a standard angle in trigonometry and its sine value should be memorized.

Step 4: Apply the reciprocal identity

Using the reciprocal identity, we find \( \csc \frac{\pi}{3} = \frac{1}{\sin \frac{\pi}{3}} = \frac{1}{\frac{\sqrt{3}}{2}} \).

Step 5: Simplify the expression

To simplify \( \frac{1}{\frac{\sqrt{3}}{2}} \), we multiply numerator and denominator by 2 to get \( \frac{2}{\sqrt{3}} \).

Step 6: Rationalize the denominator

To rationalize the denominator, multiply top and bottom by \( \sqrt{3} \): \( \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry. They allow us to relate different trigonometric functions to each other. For instance, the cosecant function, represented as \( \csc(\theta) \), is the reciprocal of the sine function. This means \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Understanding these identities can simplify complicated expressions.
Some basic trigonometric identities include:

• Sine and cosine relationship: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Reciprocal identities: \( \sec(\theta) = \frac{1}{\cos(\theta)} \), \( \csc(\theta) = \frac{1}{\sin(\theta)} \), \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• Quotient identities: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \)

These identities help you swap between different trigonometric functions easily. By learning them, you can handle tasks in trigonometry with much more confidence!

Radian Measure

Radian measure is another way to describe angles, different from degrees. While degrees divide a circle into 360 equal parts, radians divide it using the radius of the circle. One full circle is \(2\pi\) radians, which corresponds to 360 degrees. Thus, \( \pi \) radians equal 180 degrees.

To convert from degrees to radians, use the formula:

• Radians = Degrees \( \times \frac{\pi}{180} \)

For example, \(60\) degrees is:

• Radians \(= 60 \times \frac{\pi}{180} = \frac{\pi}{3} \)

Using radians makes calculations with trigonometric functions more natural in calculus. Understanding the conversion between degrees and radians is essential, especially since trigonometric functions often use radians as input.

Rationalizing the Denominator

Rationalizing the denominator is a process that involves eliminating radicals, like square roots, from the bottom of a fraction. This technique simplifies expressions and makes them easier to understand or use in further calculations.
For instance, when calculating \( \csc \frac{\pi}{3} \), we arrive at the fraction \( \frac{2}{\sqrt{3}} \). To rationalize, multiply both the numerator and the denominator by \( \sqrt{3} \) to get:

• \( \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \)

This process does not change the overall value, but it makes the expression easier to interpret. Rationalizing the denominator is particularly useful in algebra and trigonometry, ensuring that expressions can consistently be compared or integrated. By practicing this skill, you'll enhance your mathematical toolkit.