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Chapter 2: Problem 33

Complete the following tables using exact values. $$ \begin{array}{ccc} \hline x & \sin x & \csc x \\ \hline 0^{\circ} & 0 \\ 30^{\circ} & \frac{1}{2} \\ 45^{\circ} & \frac{\sqrt{2}}{2} \\ 60^{\circ} & \frac{\sqrt{3}}{2} \\ 90^{\circ} & 1 \end{array} $$

Short Answer

Expert verified
Cosecant values: 0掳 is undefined, 30掳 is 2, 45掳 is \(\sqrt{2}\), 60掳 is \(\frac{2\sqrt{3}}{3}\), 90掳 is 1.

Step by step solution

01

Understanding Sine Function Values

We begin by reviewing the given sine values for specific angles. These are standard angles in the unit circle used frequently due to their simple, exact trigonometric values. For the angle \(0^{\circ}\), the sine value is \(0\). For \(30^{\circ}\), it is \(\frac{1}{2}\). For \(45^{\circ}\), the sine value is \(\frac{\sqrt{2}}{2}\), for \(60^{\circ}\), it is \(\frac{\sqrt{3}}{2}\), and for \(90^{\circ}\), the sine value is \(1\).
02

Find Cosecant Values Using Definition

The cosecant function, \(\csc x\), is the reciprocal of the sine function. This means \(\csc x = \frac{1}{\sin x}\). We'll apply this definition to find the exact values for each angle's sine value.
03

Calculate Cosecant for 0掳

For \(x = 0^{\circ}\), \(\sin 0^{\circ} = 0\). Since cosecant is the reciprocal, \(\csc 0^{\circ} = \frac{1}{0}\), which is undefined because division by zero is not possible.
04

Calculate Cosecant for 30掳

For \(x = 30^{\circ}\), \(\sin 30^{\circ} = \frac{1}{2}\). The cosecant is \(\csc 30^{\circ} = \frac{1}{\frac{1}{2}} = 2\).
05

Calculate Cosecant for 45掳

For \(x = 45^{\circ}\), \(\sin 45^{\circ} = \frac{\sqrt{2}}{2}\). The cosecant is \(\csc 45^{\circ} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}\).
06

Calculate Cosecant for 60掳

For \(x = 60^{\circ}\), \(\sin 60^{\circ} = \frac{\sqrt{3}}{2}\). The cosecant is \(\csc 60^{\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}= \frac{2\sqrt{3}}{3}\) after rationalizing the denominator.
07

Calculate Cosecant for 90掳

For \(x = 90^{\circ}\), \(\sin 90^{\circ} = 1\). The cosecant is \(\csc 90^{\circ} = \frac{1}{1} = 1\).
08

Complete the Table

Now, we have: \(\begin{array}{ccc}\hline x & \sin x & \csc x \\hline 0^{\circ} & 0 & \text{undefined}\30^{\circ} & \frac{1}{2} & 2\45^{\circ} & \frac{\sqrt{2}}{2} & \sqrt{2}\60^{\circ} & \frac{\sqrt{3}}{2} & \frac{2\sqrt{3}}{3}\90^{\circ} & 1 & 1\end{array}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Circle
The unit circle is a fundamental concept in trigonometry that helps us understand the relationships between trigonometric functions and angles. It is a circle with a radius of one, centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle and can be represented by the coordinates \((\cos \theta, \sin \theta)\).
  • For any angle \(\theta\) in the unit circle, the x-coordinate represents the cosine value, while the y-coordinate represents the sine value.
  • The unit circle simplifies the process of finding trigonometric function values for standard angles \((0^{\circ}, 30^{\circ}, 45^{\circ}, 60^{\circ}, \,\text{and}\, 90^{\circ})\).
By visualizing these points on the circle, you can easily grasp how the sine function derives its values. The consistency and symmetry of the unit circle make it a vital tool for understanding circular motion and trigonometric functions.
Sine Function
The sine function is one of the primary trigonometric functions and is derived from the unit circle. \Given an angle \(x\), the sine function is represented as \(\sin x\), which is the y-coordinate of the corresponding point on the unit circle.
  • The sine of \(0^{\circ}\) is \(0\) because it is the starting point on the positive x-axis.
  • At \(30^{\circ}\), the y-coordinate is \(\frac{1}{2}\), representing the sine value.
  • For \(45^{\circ}\), the sine is \(\frac{\sqrt{2}}{2}\), which occurs when the angle bisects the first quadrant.
These values depict how the sine function behaves periodically as the angle increases. The sine function plays a crucial role in modeling oscillations, waves, and other periodic phenomena.
Cosecant Function
The cosecant function \(\csc x\) is the reciprocal of the sine function. It is defined as \(\csc x = \frac{1}{\sin x}\). Because it is a reciprocal, the cosecant function is undefined for angles where \(\sin x = 0\), such as \(0^{\circ}\).
  • For \(30^{\circ}, \csc 30^{\circ} = 2\) since the sine here is \(\frac{1}{2}\).
  • At \(45^{\circ}, \csc 45^{\circ} = \sqrt{2}\).
  • For \(60^{\circ}, \csc 60^{\circ} = \frac{2\sqrt{3}}{3}\), after rationalizing the reciprocal of the sine value.
The cosecant function grows significantly as the sine value approaches zero, which explains why it becomes undefined at \(x = 0^{\circ}\).
Understanding the cosecant function helps in solving trigonometric equations and analyzing signals.
Reciprocal Trigonometric Functions
Reciprocal trigonometric functions include secant, cosecant, and cotangent. They are inverses of the basic trigonometric functions: sine, cosine, and tangent, respectively. The primary purpose of reciprocal trigonometric functions is to provide solutions where typical trigonometric functions reach zero or are undefined.
  • The secant function \(\sec x\) is \(\frac{1}{\cos x}\).
  • The cosecant function \(\csc x\) is \(\frac{1}{\sin x}\).
  • The cotangent function \(\cot x\) is \(\frac{1}{\tan x}\).
Reciprocal functions are often used in calculus and trigonometric identities. They provide a complementary perspective and allow for solving equations that involve angles and circles in broader mathematical and engineering contexts.

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Most popular questions from this chapter

Danny and Stacey have gone from the swing (Example 5) to the slide at the park. The slide is inclined at an angle of \(52.0^{\circ}\). Danny weighs \(42.0\) pounds. He is sitting in a cardboard box with a piece of wax paper on the bottom. Stacey is at the top of the slide holding on to the cardboard box (Figure 23). Find the magnitude of the force Stacey must pull with, in order to keep Danny from sliding down the slide. (We are assuming that the wax paper makes the slide into a frictionless surface, so that the only force keeping Danny from sliding is the force with which Stacey pulls.)

Use the Cofunction Theorem to fill in the blanks so that each expression becomes a true statement. \(\tan 8^{\circ}=\cot\)

If the angle of elevation to the sun is \(74.3^{\circ}\) when a flagpole casts a shadow of \(22.5\) feet, what is the height of the flagpole? a. \(63.2\) feet b. \(79.5\) feet c. \(83.1\) feet d. \(80.0\) feet

If an observer positioned at the vertex of an angle views an object in the direction of the nonhorizontal side of the angle, then this side is called the ____ __ ____ of the observer.

From here on, each Problem Set will end with a series of review problems. In mathematics, it is very important to review. The more you review, the better you will understand the topics we cover and the longer you will remember them. Also, there will be times when material that seemed confusing earlier will be less confusing the second time around. The problems that follow review material we covered in Section 1.2. Find the distance between the points \((3,-2)\) and \((-1,-4)\).
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