Problem 10 - Precalculus

Question

Find the exact values of the indicated trigonometric functions using the unit circle. $$\csc \left(\frac{5 \pi}{3}\right)$$

Step-by-Step Solution

Answer

The exact value of \( \csc \left(\frac{5 \pi}{3}\right) \) is \( -\frac{2 \sqrt{3}}{3} \).

Step 1: Understand the unit circle

The unit circle is a circle with a radius of 1 centered at the origin (0,0) on the coordinate plane. Each point on the circle corresponds to an angle measured from the positive x-axis, with angles increasing counterclockwise. The x-coordinate of a point is the cosine of the angle, and the y-coordinate is the sine of the angle.

Step 2: Locate the angle on the unit circle

The angle given is \( \frac{5 \pi}{3} \) radians. To understand where this angle is located, recall that \( 2 \pi \) radians is a full circle. Thus, \( \frac{5 \pi}{3} \) can be rewritten as \( 2 \pi - \frac{ \pi }{3} \), which means it is equivalent to subtracting \( \frac{ \pi }{3} \) from \( 2 \pi \). This places the angle in the fourth quadrant.

Step 3: Identify the reference angle

The reference angle for any angle \( \theta \) in the fourth quadrant is given by \( 2 \pi - \theta \). So, we subtract \( \frac{5 \pi}{3} \) from \( 2 \pi \), which gives us the reference angle of \( \frac{ \pi }{3} \).

Step 4: Find the sine of the reference angle

For the reference angle \( \frac{ \pi }{3} \), the sine value is \( \frac{ \sqrt{3} }{2} \). However, because \( \frac{5 \pi}{3} \) is in the fourth quadrant, where sine values are negative, the sine of \( \frac{5 \pi}{3} \) is \( -\frac{ \sqrt{3} }{2} \).

Step 5: Calculate the cosecant of the angle

Cosecant is the reciprocal of sine. So, \( \csc \left(\frac{5 \pi}{3} \right) = \frac{1}{\sin \left(\frac{5 \pi}{3} \right)} \). Substituting the sine value, we get \( \csc \left(\frac{5 \pi}{3} \right) = \frac{1}{-\frac{ \sqrt{3} }{2}} = -\frac{2}{\sqrt{3}} \).

Step 6: Rationalize the denominator

To rationalize \( -\frac{2}{\sqrt{3}} \), multiply both the numerator and denominator by \( \sqrt{3} \) to get \(-\frac{2 \sqrt{3}}{3} \). Thus, the final exact value of \( \csc \left(\frac{5 \pi}{3} \right) \) is \( -\frac{2 \sqrt{3}}{3} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental mathematical functions that relate the angles of a right triangle to the ratios of its sides. These functions extend beyond triangles into the complex quadrants of the unit circle where they describe the relationships between angles and coordinates within a circle of radius one. Understanding these functions is crucial for solving trigonometric problems.

Let's break down the most common trigonometric functions:

• **Sine (\(\sin\)):** Represents the y-coordinate on the unit circle.
• **Cosine (\(\cos\)):** Represents the x-coordinate on the unit circle.
• **Tangent (\(\tan\)):** The ratio of sine to cosine, or \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
• **Cosecant (\(\csc\)):** The reciprocal of sine, expressed as \(\csc(\theta) = \frac{1}{\sin(\theta)}\).
• **Secant (\(\sec\)):** The reciprocal of cosine.
• **Cotangent (\(\cot\)):** The reciprocal of tangent.

These functions are essential for describing the periodic behavior seen in waves and oscillations, which are foundational in physics, engineering, and even music.

They are also handy tools for preparing for calculus as they provide a bridge from algebraic equations to more advanced mathematical analysis.

Reference Angle

The concept of a reference angle is a simplified way to approach angle calculations on the unit circle. A reference angle is the smallest angle formed between the terminal side of an angle and the x-axis. It is always a positive number and helps in determining the trigonometric function values for angles located in different quadrants.

The properties of reference angles can be observed in these key points:

• **Quadrant I:** The angle is its own reference angle.
• **Quadrant II:** The reference angle is \(\pi - \theta\) where \(\theta\) is the given angle.
• **Quadrant III:** The reference angle is \(\theta - \pi\).
• **Quadrant IV:** The reference angle is \(2\pi - \theta\).

In the case of \(\frac{5\pi}{3}\), the angle's reference angle is \(\frac{\pi}{3}\), calculated from quadrant IV. The reference angle is always measured to the closest x-axis line, which simplifies finding the sine and cosine values needed to subsequently compute other trigonometric functions.

Cosecant

Cosecant is one of the six primary trigonometric functions, and it represents the reciprocal of the sine function. It provides a measure of how inversely the sine relates to an angle on the unit circle. Simply put, if you have a sine value, you can find the cosecant by calculating the reciprocal.

Understanding cosecant involves a few foundational steps:

• Recall that \(\csc(\theta) = \frac{1}{\sin(\theta)}\). This indicates that wherever the sine is undefined, the cosecant will also be problematic.
• The cosecant function tends to infinity as the sine function tends towards zero.
• It is crucial to remember that because the sine function can be negative in specific quadrants, so too can the cosecant.

In the case of \(\csc\left(\frac{5\pi}{3}\right)\), starting with the known sine value for the angle derived from its reference angle provides clarity. \(\sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2}\), means the cosecant is \(\csc\left(\frac{5\pi}{3}\right) = -\frac{2}{\sqrt{3}}\), which can then be rationalized to \(-\frac{2\sqrt{3}}{3}\). This highlights how understanding the unit circle helps compute exact values for trigonometric functions.