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Chapter 1: Problem 3

(a) Complete the following table of values. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Length of arc on } \\ \text { the unit circle } \end{array} & \begin{array}{c} \text { Terminal point } \\ \text { of the arc } \end{array} & \cos (t) & \sin (t) \\ \hline 0 & (1,0) & 1 & 0 \\ \hline \frac{\pi}{2} & & & \\ \hline \pi & & & \\ \hline \frac{3 \pi}{2} & & & \\ \hline 2 \pi & & & \\ \hline \end{array} $$ (b) Complete the following table of values. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Length of arc on } \\ \text { the unit circle } \end{array} & \begin{array}{c} \text { Terminal point } \\ \text { of the arc } \end{array} & \cos (t) & \sin (t) \\ \hline 0 & (1,0) & 1 & 0 \\ \hline-\frac{\pi}{2} & & & \\ \hline-\pi & & & \\ \hline-\frac{3 \pi}{2} & & & \\ \hline-2 \pi & & & \\ \hline \end{array} $$ (c) Complete the following table of values. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Length of arc on } \\ \text { the unit circle } \end{array} & \begin{array}{c} \text { Terminal point } \\ \text { of the arc } \end{array} & \cos (t) & \sin (t) \\ \hline 2 \pi & (1,0) & 1 & 0 \\ \hline \frac{5 \pi}{2} & & & \\ \hline 3 \pi & & & \\ \hline \frac{7 \pi}{2} & & & \\ \hline 4 \pi & & & \\ \hline \end{array} $$

Short Answer

Expert verified
The short answer for completing the tables is: Table (a): $$ \begin{array}{|c|c|c|c|} \hline \text{Length of arc} & \text{Terminal point} & \cos(t) & \sin(t) \\ \hline 0 & (1,0) & 1 & 0 \\ \hline \frac{\pi}{2} & (0,1) & 0 & 1 \\ \hline \pi & (-1,0) & -1 & 0 \\ \hline \frac{3 \pi}{2} & (0,-1) & 0 & -1 \\ \hline 2 \pi & (1,0) & 1 & 0 \\ \hline \end{array} $$ Table (b): $$ \begin{array}{|c|c|c|c|} \hline \text{Length of arc} & \text{Terminal point} & \cos(t) & \sin(t) \\ \hline 0 & (1,0) & 1 & 0 \\ \hline -\frac{\pi}{2} & (0,-1) & 0 & -1 \\ \hline -\pi & (-1,0) & -1 & 0 \\ \hline -\frac{3 \pi}{2} & (0,1) & 0 & 1 \\ \hline -2 \pi & (1,0) & 1 & 0 \\ \hline \end{array} $$ Table (c): $$ \begin{array}{|c|c|c|c|} \hline \text{Length of arc} & \text{Terminal point} & \cos(t) & \sin(t) \\ \hline 2 \pi & (1,0) & 1 & 0 \\ \hline \frac{5 \pi}{2} & (0,1) & 0 & 1 \\ \hline 3 \pi & (-1,0) & -1 & 0 \\ \hline \frac{7 \pi}{2} & (0,-1) & 0 & -1 \\ \hline 4 \pi & (1,0) & 1 & 0 \\ \hline \end{array} $$

Step by step solution

01

Table (a)

To complete table (a), follow these steps for each angle measure: 1. Determine the coordinates of the terminal point on the unit circle. 2. Find the cosine and sine values corresponding to the coordinates. 1. For angle measure \(\frac{\pi}{2}\):
02

Terminal point

The terminal point is located at \((0, 1)\).
03

Cosine and sine values

The cosine value is 0 and the sine value is 1. 2. For angle measure \(\pi\):
04

Terminal point

The terminal point is located at \((-1, 0)\).
05

Cosine and sine values

The cosine value is -1 and the sine value is 0. 3. For angle measure \(\frac{3 \pi}{2}\):
06

Terminal point

The terminal point is located at \(0, -1\).
07

Cosine and sine values

The cosine value is 0 and the sine value is -1. 4. For angle measure \(2 \pi\):
08

Terminal point

The terminal point is located at \((1, 0)\).
09

Cosine and sine values

The cosine value is 1 and the sine value is 0. Now the completed table (a) looks like this: $$ \begin{array}{|c|c|c|c|} \hline \text{Length of arc} & \text{Terminal point} & \cos(t) & \sin(t) \\ \hline 0 & (1,0) & 1 & 0 \\ \hline \frac{\pi}{2} & (0,1) & 0 & 1 \\ \hline \pi & (-1,0) & -1 & 0 \\ \hline \frac{3 \pi}{2} & (0,-1) & 0 & -1 \\ \hline 2 \pi & (1,0) & 1 & 0 \\ \hline \end{array} $$
10

Table (b)

Follow the same steps for table (b): 1. For angle measure -\(\frac{\pi}{2}\):
11

Terminal point

The terminal point is located at \((0, -1)\).
12

Cosine and sine values

The cosine value is 0 and the sine value is -1. 2. For angle measure -\(\pi\):
13

Terminal point

The terminal point is located at \((-1, 0)\).
14

Cosine and sine values

The cosine value is -1 and the sine value is 0. 3. For angle measure -\(\frac{3 \pi}{2}\):
15

Terminal point

The terminal point is located at \((0, 1)\).
16

Cosine and sine values

The cosine value is 0 and the sine value is 1. 4. For angle measure -\(2 \pi\):
17

Terminal point

The terminal point is located at \((1, 0)\).
18

Cosine and sine values

The cosine value is 1 and the sine value is 0. Now the completed table (b) looks like this: $$ \begin{array}{|c|c|c|c|} \hline \text{Length of arc} & \text{Terminal point} & \cos(t) & \sin(t) \\ \hline 0 & (1,0) & 1 & 0 \\ \hline -\frac{\pi}{2} & (0,-1) & 0 & -1 \\ \hline -\pi & (-1,0) & -1 & 0 \\ \hline -\frac{3 \pi}{2} & (0,1) & 0 & 1 \\ \hline -2 \pi & (1,0) & 1 & 0 \\ \hline \end{array} $$
19

Table (c)

Follow the same steps for table (c): 1. For angle measure \(\frac{5 \pi}{2}\):
20

Terminal point

The terminal point is located at \((0, 1)\).
21

Cosine and sine values

The cosine value is 0 and the sine value is 1. 2. For angle measure \(3 \pi\):
22

Terminal point

The terminal point is located at \((-1, 0)\).
23

Cosine and sine values

The cosine value is -1 and the sine value is 0. 3. For angle measure \(\frac{7 \pi}{2}\):
24

Terminal point

The terminal point is located at \((0, -1)\).
25

Cosine and sine values

The cosine value is 0 and the sine value is -1. 4. For angle measure \(4 \pi\):
26

Terminal point

The terminal point is located at \((1, 0)\).
27

Cosine and sine values

The cosine value is 1 and the sine value is 0. Now the completed table (c) looks like this: $$ \begin{array}{|c|c|c|c|} \hline \text{Length of arc} & \text{Terminal point} & \cos(t) & \sin(t) \\ \hline 2 \pi & (1,0) & 1 & 0 \\ \hline \frac{5 \pi}{2} & (0,1) & 0 & 1 \\ \hline 3 \pi & (-1,0) & -1 & 0 \\ \hline \frac{7 \pi}{2} & (0,-1) & 0 & -1 \\ \hline 4 \pi & (1,0) & 1 & 0 \\ \hline \end{array} $$

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Key Concepts

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

Trigonometric Functions
Trigonometric functions are essential tools in mathematics that relate angles to side lengths in right-angled triangles. On the unit circle, these functions help us understand how angles correspond to coordinates on the edge of the circle. The two primary trigonometric functions you will encounter are sine (\( \sin \)) and cosine (\( \cos \)). These functions allow us to calculate the x and y coordinates of a point on the circle based on a given angle. This relationship is foundational in understanding how angles translate to positions on the unit circle.

Key trigonometric functions include:
  • **Sine (\( \sin \)):** Defines the y-coordinate of a point on the unit circle.
  • **Cosine (\( \cos \)):** Defines the x-coordinate of a point on the unit circle.
  • **Tangent (\( \tan \)):** Calculated as \( \sin/\cos \) and not typically a focus on the unit circle, but still important in broader trigonometry.
Understanding these functions helps you describe circle-related concepts and solve problems involving angles and lengths.
Sine and Cosine
Sine and cosine are the most important trigonometric functions used on the unit circle. Their values provide critical insights into the relationship between angles and their respective coordinates on the circle.

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. For any angle \( t \), the sine function gives the y-coordinate, while the cosine function provides the x-coordinate of the vertex reached by angle \( t \).
  • For the angle \( t = 0 \), the coordinates are \((1, 0)\), meaning \( \cos(0) = 1 \), and \( \sin(0) = 0 \).
  • An angle of \( \frac{\pi}{2} \) results in coordinates \((0, 1)\) because at this point, \( \cos(\frac{\pi}{2}) = 0 \) and \( \sin(\frac{\pi}{2}) = 1 \).
  • As the angles progress to \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \), the coordinates cycle through \((-1, 0)\), \((0, -1)\), and back to \((1, 0)\), reflecting the periodic nature of these functions.
Sine and cosine functions are periodic, and this pattern repeats every \( 2\pi \). This periodicity is why these functions are so critical in physics, engineering, and signal analysis.
Angle Measures
Understanding angle measures is crucial when working with the unit circle and trigonometric functions. Angles can be measured in different units, such as degrees or radians, with radians being more commonly used in trigonometry and calculus.
  • **Radians:** This is the standard unit of angular measure used in many areas of mathematics. One full circle around the unit circle corresponds to an angle of \( 2\pi \) radians.
  • **Degrees:** Often used in practical applications, where a full circle equals 360 degrees.
When completing tables like the ones in the exercise, it is important to convert angles to radians if they are given in degrees, as this is necessary for accurately determining the terminal point on the unit circle and calculating sine and cosine values.

By using the concept of angle measures, you can easily understand rotations on the circle. So, knowing how each angle aligns with the points on the unit circle helps predict the corresponding sine and cosine values accurately.

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Most popular questions from this chapter

A person is riding on a Ferris wheel that takes 28 seconds to make a complete revolution. Her seat is 25 feet from the axle of the wheel. (a) What is her angular velocity in revolutions per minute? Radians per minute? Degrees per minute? (b) What is her linear velocity? (c) Which of the quantities angular velocity and linear velocity change if the person's seat was 20 feet from the axle instead of 25 feet? Compute the new value for any value that changes. Explain why each value changes or does not change.

Use a calculator to determine four-digit decimal approximations for each of the following. (a) \(\csc (1)\) (b) \(\tan \left(\frac{12 \pi}{5}\right)\) (c) \(\cot (5)\) (d) \(\sec \left(\frac{13 \pi}{8}\right)\) (e) \(\sin ^{2}(5.5)\) (f) \(1+\tan ^{2}(2)\) (g) \(\sec ^{2}(2)\)

Draw the following arcs on the unit circle. (a) The arc that is determined by the interval \(\left[0, \frac{\pi}{6}\right]\) on the number line. (b) The arc that is determined by the interval \(\left[0, \frac{7 \pi}{6}\right]\) on the number line. (c) The arc that is determined by the interval \(\left[0,-\frac{\pi}{3}\right]\) on the number line. (d) The arc that is determined by the interval \(\left[0,-\frac{4 \pi}{5}\right]\) on the number line.

A compact disc (CD) has a diameter of 12 centimeters (cm). Suppose that the CD is in a CD-player and is rotating at 225 revolutions per minute. What is the angular velocity of the CD (in radians per second) and what is the linear velocity of a point on the edge of the CD?

This exercise provides an alternate method for determining the exact values of \(\cos \left(\frac{\pi}{6}\right)\) and \(\sin \left(\frac{\pi}{6}\right)\). The diagram to the right shows the terminal point \(P(x, y)\) for an arc of length \(t=\frac{\pi}{6}\) on the unit circle. The points \(A(1,0)\), \(B(0,1),\) and \(C(x,-y)\) are also shown. Notice that \(B\) is the terminal point of the \(\operatorname{arc} t=\frac{\pi}{2},\) and \(C\) is the terminal point of the arc \(t=-\frac{\pi}{6}\). We now notice that the length of the arc from \(P\) to \(B\) is $$ \frac{\pi}{2}-\frac{\pi}{6}=\frac{\pi}{3} $$ In addition, the length of the arc from \(C\) to \(P\) is $$ \frac{\pi}{6}-\frac{-\pi}{6}=\frac{\pi}{3} $$ This means that the distance from \(P\) to \(B\) is equal to the distance from \(C\) to \(P\) (a) Use the distance formula to write a formula (in terms of \(x\) and \(y\) ) for the distance from \(P\) to \(B\). (b) Use the distance formula to write a formula (in terms of \(x\) and \(y\) ) for the distance from \(C\) to \(P\). (c) Set the distances from (a) and (b) equal to each other and solve the resulting equation for \(y\). To do this, begin by squaring both sides of the equation. In order to solve for \(y\), it may be necessary to use the fact that \(x^{2}+y^{2}=1\) (d) Use the value for \(y\) in (c) and the fact that \(x^{2}+y^{2}=1\) to determine the value for \(x\). Explain why this proves that $$\cos \left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2} \text { and } \sin \left(\frac{\pi}{3}\right)=\frac{1}{2}$$.
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