Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Begin by reproducing the graph in RiGuRE as. Keep in mind that for each of the four points labeled in the figure, \(r=1 .\) For each quadrantal angle, identify the appropriate values of \(x, y,\) and \(r\) to find the indicated function value. If it is undefined, say so. Check your answers with a calculator in degree mode. $$\csc 270^{\circ}$$

Step-by-Step Solution

Answer

The value of \( \csc(270^{\circ}) \) is -1.

Step 1: Understanding the context

The problem asks us to find the value of the cosecant function for an angle of 270 degrees. Cosecant is the reciprocal of the sine function. Given the angle in standard position, we need to find the corresponding point on the unit circle.

Step 2: Identifying the coordinates

For the angle 270 degrees on the unit circle, the point is located at the coordinates \( (x, y) = (0, -1) \). At any point on the unit circle, the value of \( r \) (the radius) is 1.

Step 3: Finding the sine

The sine of the angle 270 degrees is determined by the y-coordinate of the unit circle point. Therefore, \( \sin(270^{\circ}) = -1 \).

Step 4: Calculating the cosecant

The cosecant of an angle is the reciprocal of sine. Thus, \( \csc(270^{\circ}) = \frac{1}{\sin(270^{\circ})} \). Substituting the value of sine, we get \( \csc(270^{\circ}) = \frac{-1}{1} = -1 \).

Step 5: Verifying with calculator

Using a calculator in degree mode, we can confirm the calculation: Entering the command for cosecant of 270 degrees, you will get a value of -1, confirming our result.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry and is crucial for understanding trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate plane.
The equation for this circle is \[ x^2 + y^2 = 1 \]. This circle helps to define the sine, cosine, and other trigonometric functions. Each point \((x, y)\) on the unit circle represents the cosine and sine of an angle \( heta \), respectively. This means for any angle \( heta \), we can associate \( ext{cos}( heta) = x \) and \( ext{sin}( heta) = y \).

• The unit circle simplifies calculations, as it allows you to understand trigonometric values just by finding the coordinates of a point on the circle.
• Angles can be measured in radians or degrees, but the unit circle commonly uses degrees for ease in basic learning.
• The circle is divided into 4 quadrants, helping to understand the sign of trigonometric functions based on the quadrant in which the angle falls.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are opposite or inverse functions of the basic trigonometric functions. These include cosecant (csc), secant (sec), and cotangent (cot). Each of these reciprocal functions offers a different perspective or transformation of the original functions and follows the unit circle's rules.
For example:

• Cosecant: \( ext{csc}( heta) = rac{1}{ ext{sin}( heta)} \)
• Secant: \( ext{sec}( heta) = rac{1}{ ext{cos}( heta)} \)
• Cotangent: \( ext{cot}( heta) = rac{1}{ ext{tan}( heta)} \)

These functions are essential when trigonometric functions approach zero since the reciprocal will then point towards infinity, making the original function undefined at certain angles. For instance, \( ext{csc}(270^{ ext{o}}) \) is the reciprocal of \( ext{sin}(270^{ ext{o}}) \), which equals \(-1\), the final calculation giving \(-1\). This illustrates that the values and behavior of functions on the unit circle also translate directly to their reciprocals, maintaining the identities across transformations.

Quadrantal Angles

Quadrantal angles are those angles located on the axes of the coordinate system and are vital in simplifying trigonometric computations. These angles include 0°, 90°, 180°, 270°, and 360°, marking the completion of a full revolution around the circle. Each quadrantal angle corresponds to specific \((x, y)\) coordinates on the unit circle where either the sine or cosine equals zero, simplifying these instances.

• For 0° and 360°: point is \((1, 0)\)
• For 90°: point is \((0, 1)\)
• For 180°: point is \((-1, 0)\)
• For 270°: point is \((0, -1)\)

These angles not only help define \(x, y, ext{and } r\) accurately but also reveal the limits of trigonometric functions at essential points. In solving problems, they guide determining the signs of trigonometric ratios, as seen in calculating \( ext{csc}(270^{ ext{o}}) \), where the unit circle and understanding of quadrantal angles converge neatly.