Problem 338 - Precalculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Expression

The task requires us to find the exact value of \( \csc \frac{\pi}{3} \).The cosecant function, \( \csc(x) \), is equal to \( \frac{1}{\sin(x)} \). Thus, first, we need to evaluate \( \sin \frac{\pi}{3} \).

Step 2: Evaluate the Sine Value

Recall that \( \frac{\pi}{3} \) radians is equivalent to 60 degrees.The sine of 60 degrees (or \( \frac{\pi}{3} \) radians) is a known value.\[ \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \]

Step 3: Use the Definition of Cosecant

Using the relationship \( \csc(x) = \frac{1}{\sin(x)} \), we compute the value:\[ \csc \frac{\pi}{3} = \frac{1}{\sin \frac{\pi}{3}} = \frac{1}{\frac{\sqrt{3}}{2}} \]

Step 4: Simplify the Expression

To simplify \( \frac{1}{\frac{\sqrt{3}}{2}} \), multiply the numerator and denominator by 2 to eliminate the fraction in the denominator:\[ \csc \frac{\pi}{3} = \frac{2}{\sqrt{3}} \]Further, simplify by rationalizing the denominator:\[ \csc \frac{\pi}{3} = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

Key Concepts

Cosecant Function

The cosecant function, abbreviated as \( \csc(x) \), is one of the key trigonometric functions. It is essentially the reciprocal of the sine function. This means that for any angle \( x \), the relationship is given by:

• \( \csc(x) = \frac{1}{\sin(x)} \)

This relationship implies that where the sine function is undefined (such as angles where sine equals zero), the cosecant function will also be undefined. Cosecant primarily deals with ratios in right-angled triangles, tying closely to trigonometric identities and solutions of trigonometric equations. Understanding the reciprocal nature helps in solving trigonometric problems easily.

Sine Function

The sine function, denoted as \( \sin(x) \), is a fundamental trigonometric function that represents the ratio of the side opposite the angle to the hypotenuse in a right-angled triangle. Mathematically, for an angle \( x \) in a right triangle, the sine function is defined as:

• \( \sin(x) = \frac{opposite}{hypotenuse} \)

Sine is an important aspect when dealing with periodic phenomena, such as sound waves and tides. In a unit circle, the sine of an angle corresponds to the y-coordinate of the point where the terminal side of the angle intersects the circle. This fundamental understanding helps in evaluating various trigonometric expressions and solving equations involving sine.

Radian Measure

Radian measure is a method of measuring angles, which provides an alternative to degrees. Unlike a degree, which divides a circle into 360 parts, a radian is based on the radius of a circle. An angle in radians is the length of the arc on the unit circle subtended by the angle. The relationship between radians and degrees is given by:

• \( \pi \) radians = 180 degrees

This means that to convert from radians to degrees, you multiply by \( \frac{180}{\pi} \), and to convert from degrees to radians, you multiply by \( \frac{\pi}{180} \). For example, \( \frac{\pi}{3} \) radians is equivalent to 60 degrees, a conversion that is frequently used in trigonometry to simplify problems.

Exact Trigonometric Values

Exact trigonometric values refer to specific, often non-decimal values that are fundamental in trigonometry. These are precise values derived mathematically, rather than approximations calculated on a calculator. Common angles where these values are found include 0, 30, 45, 60, and 90 degrees or their radian equivalents: \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \).For example, at \( \frac{\pi}{3} \) radians (or 60 degrees), the exact sine value is \( \frac{\sqrt{3}}{2} \). To find the cosecant, you take the reciprocal of the sine value resulting in \( \frac{2\sqrt{3}}{3} \). Knowing these values by heart is helpful in quickly solving trigonometric problems without needing a calculator.