Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Concept explainers
Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
Question
![**Title: Calculating Trigonometric Values Using the Unit Circle**
The terminal ray of angle \(\theta\) in standard position passes through the point \((-0.8, -0.6)\) on the unit circle, as shown below. Remember to put your calculator in degrees.
![Unit Circle Diagram]
*A unit circle with a terminal ray passing through the point \((-0.8, -0.6)\) is shown. The angle \(\theta\) is indicated by the arc connecting the positive x-axis to the terminal ray.*
### Questions
**a) What is the value of \(\sin{\theta}\)?**
**b) What is the value of \(\cos{\theta}\)?**
**c) What is the value of \(\theta\) (round to the nearest hundredth; two decimal places)?**
### Explanation of the Diagram
In this diagram, the unit circle is plotted on a coordinate plane with the center at the origin \((0,0)\). The circle has a radius of 1. The terminal ray forms an angle \(\theta\) with the positive x-axis and passes through a point on the circle. The coordinates of this point are \((-0.8, -0.6)\), which are essential for calculating the trigonometric values.
To solve the above questions, consider the following:
- The sine of the angle \(\theta\) corresponds to the y-coordinate of the given point.
- The cosine of the angle \(\theta\) is the x-coordinate of the given point.
- To find the angle \(\theta\), we use the inverse trigonometric functions and the coordinates of the point.
### Instructions
Answer the questions below and UPLOAD or show your work in question 2.
**a) What is the value of \(\sin{\theta}\)?**
\[ \sin{\theta} = \_\_\_\_ \]
**b) What is the value of \(\cos{\theta}\)?**
\[ \cos{\theta} = \_\_\_\_ \]
**c) What is the value of \(\theta\) (round to the nearest hundredth; two decimal places)?**
\[ \theta = \_\_\_.\_\_^\circ \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F945c8a79-affc-4bd6-bac5-afc665d5b633%2Fd48975fc-7d04-4e17-9b3b-4e61790e9697%2Fkdhcnx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Calculating Trigonometric Values Using the Unit Circle**
The terminal ray of angle \(\theta\) in standard position passes through the point \((-0.8, -0.6)\) on the unit circle, as shown below. Remember to put your calculator in degrees.
![Unit Circle Diagram]
*A unit circle with a terminal ray passing through the point \((-0.8, -0.6)\) is shown. The angle \(\theta\) is indicated by the arc connecting the positive x-axis to the terminal ray.*
### Questions
**a) What is the value of \(\sin{\theta}\)?**
**b) What is the value of \(\cos{\theta}\)?**
**c) What is the value of \(\theta\) (round to the nearest hundredth; two decimal places)?**
### Explanation of the Diagram
In this diagram, the unit circle is plotted on a coordinate plane with the center at the origin \((0,0)\). The circle has a radius of 1. The terminal ray forms an angle \(\theta\) with the positive x-axis and passes through a point on the circle. The coordinates of this point are \((-0.8, -0.6)\), which are essential for calculating the trigonometric values.
To solve the above questions, consider the following:
- The sine of the angle \(\theta\) corresponds to the y-coordinate of the given point.
- The cosine of the angle \(\theta\) is the x-coordinate of the given point.
- To find the angle \(\theta\), we use the inverse trigonometric functions and the coordinates of the point.
### Instructions
Answer the questions below and UPLOAD or show your work in question 2.
**a) What is the value of \(\sin{\theta}\)?**
\[ \sin{\theta} = \_\_\_\_ \]
**b) What is the value of \(\cos{\theta}\)?**
\[ \cos{\theta} = \_\_\_\_ \]
**c) What is the value of \(\theta\) (round to the nearest hundredth; two decimal places)?**
\[ \theta = \_\_\_.\_\_^\circ \]
![### Understanding Trigonometric Functions on a Circle
For the circle defined by the equation \( x^2 + y^2 = r^2 \):
**Circumference:**
\[ \text{Circumference} = 2\pi r \]
**Defining the Angle and Trigonometric Ratios:**
If \(\theta\) is an angle in standard position, where its terminal ray passes through point \(P(x, y)\) on the circle, then the following trigonometric functions are defined as:
1. **Sine:**
\[
\sin \theta = \frac{y}{r}
\]
2. **Cosine:**
\[
\cos \theta = \frac{x}{r}
\]
3. **Tangent:**
\[
\tan \theta = \frac{y}{x}
\]
**Pythagorean Identity:**
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
**Angle Measurement Conversion:**
\[
180 \text{ degrees} = \pi \text{ radians}
\]
### Explanation of the Diagram
The accompanying diagram depicts a circle with radius \(r\) centered at the origin of a coordinate plane. In this diagram:
- Point \(P(x, y)\) is a point on the circumference of the circle.
- The angle \(\theta\) is shown in standard position, meaning its initial side is along the positive x-axis and its terminal side passes through point \(P\).
- The circle intersects the coordinate axes at \((r, 0)\), \((0, r)\), \((-r, 0)\), and \((0, -r)\).
This visual representation helps in understanding how the coordinates \(x\) and \(y\) of point \(P\) relate to the trigonometric functions sine, cosine, and tangent for a given angle \(\theta\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F945c8a79-affc-4bd6-bac5-afc665d5b633%2Fd48975fc-7d04-4e17-9b3b-4e61790e9697%2Fc7nsjrj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Trigonometric Functions on a Circle
For the circle defined by the equation \( x^2 + y^2 = r^2 \):
**Circumference:**
\[ \text{Circumference} = 2\pi r \]
**Defining the Angle and Trigonometric Ratios:**
If \(\theta\) is an angle in standard position, where its terminal ray passes through point \(P(x, y)\) on the circle, then the following trigonometric functions are defined as:
1. **Sine:**
\[
\sin \theta = \frac{y}{r}
\]
2. **Cosine:**
\[
\cos \theta = \frac{x}{r}
\]
3. **Tangent:**
\[
\tan \theta = \frac{y}{x}
\]
**Pythagorean Identity:**
\[
\sin^2 \theta + \cos^2 \theta = 1
\]
**Angle Measurement Conversion:**
\[
180 \text{ degrees} = \pi \text{ radians}
\]
### Explanation of the Diagram
The accompanying diagram depicts a circle with radius \(r\) centered at the origin of a coordinate plane. In this diagram:
- Point \(P(x, y)\) is a point on the circumference of the circle.
- The angle \(\theta\) is shown in standard position, meaning its initial side is along the positive x-axis and its terminal side passes through point \(P\).
- The circle intersects the coordinate axes at \((r, 0)\), \((0, r)\), \((-r, 0)\), and \((0, -r)\).
This visual representation helps in understanding how the coordinates \(x\) and \(y\) of point \(P\) relate to the trigonometric functions sine, cosine, and tangent for a given angle \(\theta\).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 4 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra and Trigonometry (6th Edition)
Algebra
ISBN:
9780134463216
Author:
Robert F. Blitzer
Publisher:
PEARSON
Contemporary Abstract Algebra
Algebra
ISBN:
9781305657960
Author:
Joseph Gallian
Publisher:
Cengage Learning
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra And Trigonometry (11th Edition)
Algebra
ISBN:
9780135163078
Author:
Michael Sullivan
Publisher:
PEARSON
Introduction to Linear Algebra, Fifth Edition
Algebra
ISBN:
9780980232776
Author:
Gilbert Strang
Publisher:
Wellesley-Cambridge Press
College Algebra (Collegiate Math)
Algebra
ISBN:
9780077836344
Author:
Julie Miller, Donna Gerken
Publisher:
McGraw-Hill Education