Consider the two circles a(t) = (cos(t), sin(t)), B(t) = (2+2 cos(t), 1+2 sin(t)). where t e [0, 27]. In this problem, we will show that the two circles meet at a right-angle at the point Q, as in the below diagram. -34 25 15 05 15 -1 -0.5 10 0.5 1.5 2.5 3.5 4.5 -0.5 (a) Your friend Tonga is trying to find the exact Cartesian coordinates of Q. But she ends up proving that the two circles never actually intersect. Here is her argument: To find the intersection points, we set a(t) = B(t) and solve for t. This results in two equations: cos(t) = 2+2 cos(t) and sin(t) = 1+2 sin(t). But the first equation is equivalent to cos(t) = -2, which has no solution! This means that there is no value of t for which a(t) = B(t), which means that the two circles never intersect. What's wrong with Tonga's argument? 2,
Consider the two circles a(t) = (cos(t), sin(t)), B(t) = (2+2 cos(t), 1+2 sin(t)). where t e [0, 27]. In this problem, we will show that the two circles meet at a right-angle at the point Q, as in the below diagram. -34 25 15 05 15 -1 -0.5 10 0.5 1.5 2.5 3.5 4.5 -0.5 (a) Your friend Tonga is trying to find the exact Cartesian coordinates of Q. But she ends up proving that the two circles never actually intersect. Here is her argument: To find the intersection points, we set a(t) = B(t) and solve for t. This results in two equations: cos(t) = 2+2 cos(t) and sin(t) = 1+2 sin(t). But the first equation is equivalent to cos(t) = -2, which has no solution! This means that there is no value of t for which a(t) = B(t), which means that the two circles never intersect. What's wrong with Tonga's argument? 2,
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 59E
Related questions
Question
![Consider the two circles
a(t) = (cos(t), sin(t)),
B(t) = (2+2 cos(t), 1+2 sin(t)).
where t e [0, 27]. In this problem, we will show that the two circles meet at a right-angle at the point
Q, as in the below diagram.
3-
25
15
05
15
-0.5
10
0.5
1.5
2.5
3.5
4.5
-05
(a) Your friend Tonga is trying to find the exact Cartesian coordinates of Q. But she ends up
proving that the two circles never actually intersect. Here is her argument:
To find the intersection points, we set a(t) = B(t) and solve for t. This results in two
equations: cos(t) = 2+2 cos(t) and sin(t) = 1+2 sin(t). But the first equation is equivalent
to cos(t) = -2, which has no solution! This means that there is no value of t for which
a(t) = B(t), which means that the two circles never intersect.
What's wrong with Tonga's argument?
(b) Find the exact Cartesian coordinates of Q by solving the two simultaneous equations r? +y? = 1
and (z – 2)? + (y – 1)² = 4.
(c) At what time does a reach Q? At what time does 3 reach Q?
(Hint: arctan is your friend and your enemy. It only outputs angles between –7/2 and T/2.
But the two angles you're looking for are not in that range .)
(d) Compute the tangent vectors of a and B at point Q and show that they are orthogonal.
(Fun Fact: Since the two circles meet at a right-angle, we say that they intersect transversally.)
2,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faba73b71-75cf-41e8-b53e-fd46ec14d4a7%2F17ab7f64-38dc-461a-b40b-13c1f2ffc75d%2Fy8oa0lo_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the two circles
a(t) = (cos(t), sin(t)),
B(t) = (2+2 cos(t), 1+2 sin(t)).
where t e [0, 27]. In this problem, we will show that the two circles meet at a right-angle at the point
Q, as in the below diagram.
3-
25
15
05
15
-0.5
10
0.5
1.5
2.5
3.5
4.5
-05
(a) Your friend Tonga is trying to find the exact Cartesian coordinates of Q. But she ends up
proving that the two circles never actually intersect. Here is her argument:
To find the intersection points, we set a(t) = B(t) and solve for t. This results in two
equations: cos(t) = 2+2 cos(t) and sin(t) = 1+2 sin(t). But the first equation is equivalent
to cos(t) = -2, which has no solution! This means that there is no value of t for which
a(t) = B(t), which means that the two circles never intersect.
What's wrong with Tonga's argument?
(b) Find the exact Cartesian coordinates of Q by solving the two simultaneous equations r? +y? = 1
and (z – 2)? + (y – 1)² = 4.
(c) At what time does a reach Q? At what time does 3 reach Q?
(Hint: arctan is your friend and your enemy. It only outputs angles between –7/2 and T/2.
But the two angles you're looking for are not in that range .)
(d) Compute the tangent vectors of a and B at point Q and show that they are orthogonal.
(Fun Fact: Since the two circles meet at a right-angle, we say that they intersect transversally.)
2,
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
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 3 steps with 3 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Algebra and Trigonometry (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305071742/9781305071742_smallCoverImage.gif)
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
![Trigonometry (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781337278461/9781337278461_smallCoverImage.gif)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
![Algebra and Trigonometry (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781305071742/9781305071742_smallCoverImage.gif)
Algebra and Trigonometry (MindTap Course List)
Algebra
ISBN:
9781305071742
Author:
James Stewart, Lothar Redlin, Saleem Watson
Publisher:
Cengage Learning
![Trigonometry (MindTap Course List)](https://www.bartleby.com/isbn_cover_images/9781337278461/9781337278461_smallCoverImage.gif)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning