There's already a formula for this question. But I don't understand it. Can we not use the arc sin to find the angles because this isn't a right triangle? I've attached 2 images. the first is the question. the second is the formula we were provided for this problem on the site. On the second photo. please 1.) explain why we're using 2ABCos theta and 2.) where the 128 before Cos theta went. Thank you
There's already a formula for this question. But I don't understand it. Can we not use the arc sin to find the angles because this isn't a right triangle? I've attached 2 images. the first is the question. the second is the formula we were provided for this problem on the site. On the second photo. please 1.) explain why we're using 2ABCos theta and 2.) where the 128 before Cos theta went. Thank you
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter5: Trigonometric Functions: Right Triangle Approach
Section5.1: Angle Measure
Problem 4E: Object A is travelling along a circle of radius 2, and Object B is travelling along a circle of...
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There's already a formula for this question. But I don't understand it. Can we not use the arc sin to find the
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### Understanding Vector Magnitudes and Angles
Vectors **A** and **B** both have a magnitude of 8.0. If the sum of these two vectors is 10, calculate the cosine of the angle between them.
#### Formula:
\[ R^2 = A^2 + B^2 + 2AB \cos\theta \]
### Steps:
1. **Substitute the values:**
Given:
\[ R = 10, A = 8, B = 8 \]
Plugging them into the formula:
\[ 10^2 = 8^2 + 8^2 + 2 \cdot 8 \cdot 8 \cdot \cos \theta \]
\[ 100 = 64 + 64 + 128 \cos \theta \]
2. **Solve for \( \cos \theta \):**
Simplify the equation:
\[ 100 = 128 + 128 \cos \theta \]
Subtract 128 from both sides:
\[ \frac{100}{128} = 1 + \cos \theta \]
Isolate \( \cos \theta \):
\[ \cos \theta = \frac{100}{128} - 1 \]
\[ \cos \theta = \frac{100 - 128}{128} \]
\[ \cos \theta = \frac{-28}{128} \]
Simplify the fraction:
\[ \cos \theta = \frac{-7}{32} \]
### Conclusion:
The cosine of the angle \( \theta \) between vectors **A** and **B** is \( \cos \theta = -\frac{7}{32} \).
---
#### Visualization:
While no graphs or diagrams are provided here, understanding the relationship between vectors visually can greatly aid in comprehension. In this problem, visualizing the vectors originating from a common point and forming an angle \( \theta \) provides insight into their interaction and resultant vector magnitude.
For further assistance with vectors and angles, feel free to explore additional resources or ask questions through our educational platform.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d4854d9-aadf-4fe0-b1f6-bd622f90aeab%2F0d901358-7fa3-496c-b21c-e180c9bbdb51%2F0jy8v1k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Understanding Vector Magnitudes and Angles
Vectors **A** and **B** both have a magnitude of 8.0. If the sum of these two vectors is 10, calculate the cosine of the angle between them.
#### Formula:
\[ R^2 = A^2 + B^2 + 2AB \cos\theta \]
### Steps:
1. **Substitute the values:**
Given:
\[ R = 10, A = 8, B = 8 \]
Plugging them into the formula:
\[ 10^2 = 8^2 + 8^2 + 2 \cdot 8 \cdot 8 \cdot \cos \theta \]
\[ 100 = 64 + 64 + 128 \cos \theta \]
2. **Solve for \( \cos \theta \):**
Simplify the equation:
\[ 100 = 128 + 128 \cos \theta \]
Subtract 128 from both sides:
\[ \frac{100}{128} = 1 + \cos \theta \]
Isolate \( \cos \theta \):
\[ \cos \theta = \frac{100}{128} - 1 \]
\[ \cos \theta = \frac{100 - 128}{128} \]
\[ \cos \theta = \frac{-28}{128} \]
Simplify the fraction:
\[ \cos \theta = \frac{-7}{32} \]
### Conclusion:
The cosine of the angle \( \theta \) between vectors **A** and **B** is \( \cos \theta = -\frac{7}{32} \).
---
#### Visualization:
While no graphs or diagrams are provided here, understanding the relationship between vectors visually can greatly aid in comprehension. In this problem, visualizing the vectors originating from a common point and forming an angle \( \theta \) provides insight into their interaction and resultant vector magnitude.
For further assistance with vectors and angles, feel free to explore additional resources or ask questions through our educational platform.
Transcribed Image Text:### Trigonometry Practice Problem
#### 1.4 Trigonometry
**Problem Statement:**
Vectors \( \vec{A} \) and \( \vec{B} \) both have a magnitude of 8.0. If their resultant has a magnitude of 10, what is the angle between them?
**Options:**
- 103°
- 110°
- 77°
- 26°
- 90°
**Actions:**
- Button to save the problem for later evaluation ("Save for Later").
- Button to submit the answer.
**Tools and Resources:**
This problem involves the use of vector addition and trigonometric principles to determine the angle between the two vectors. Students may want to apply the Law of Cosines as part of their solution process.
The question is accessible under the "Adaptive Practice" section of the platform, ensuring students can attempt problems tailored to their skill level. Additional resources and help may be available on the platform to assist in understanding and solving this trigonometry problem.
**Note:**
This section is designed to test the understanding of vector magnitudes and angles, crucial for mastering the concepts of trigonometry in physics and mathematics.
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