(c) Let 0, w', and z' be the three vertices of the rotated triangle. Show that Re[z] and Im[2] = zw+Zw zw-Iw 2 w (d) Show that the area of the rotated triangle is. (Since rotation doesn't change the area, your formula also gives the area of triangle OAC. (e) Let OA'C' denote the rotated triangle. Express the cosine of angle ZA'OC" in terms of w and z. (f) Let |OA', OC', and A'C' denote the lengths of the three sides of the rotated triangle. Use complex arithmetic with w and z to prove the law of cosines: |A'C'² = |OA|²+|OC|22|OA||OC| cos(ZA'OC'). (Since rotation does not change lengths or angles, you have also proved the law of cosines for the original triangle OAC.)
(c) Let 0, w', and z' be the three vertices of the rotated triangle. Show that Re[z] and Im[2] = zw+Zw zw-Iw 2 w (d) Show that the area of the rotated triangle is. (Since rotation doesn't change the area, your formula also gives the area of triangle OAC. (e) Let OA'C' denote the rotated triangle. Express the cosine of angle ZA'OC" in terms of w and z. (f) Let |OA', OC', and A'C' denote the lengths of the three sides of the rotated triangle. Use complex arithmetic with w and z to prove the law of cosines: |A'C'² = |OA|²+|OC|22|OA||OC| cos(ZA'OC'). (Since rotation does not change lengths or angles, you have also proved the law of cosines for the original triangle OAC.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please do part C, D,E, and F and please show steps and explain
For C: replace
Im[z'] = (\bar{z} w -z \bar{w}) / (2|w|)
with
Im[z'] = -i(\bar{z} w -z \bar{w}) / (2|w|)
(multiply the RHS by -i).
![Exercise 4.4.15. Consider a plane with Cartesian coordinates. Let O be
the point (0,0), let A be the point (a, b), and let C be the point (c,d). Also,
let w = a + bi and z = c+di. We may consider the three complex numbers
0, w, z as representing the vertices of triangle OAC.
(A word to the wise: drawing a picture can be extremely helpful.)
(a) Express the lengths of the three sides of the triangle in terms of u and
z. For example, the length of side OA is |w|.
(b) Show that multiplying 0, w, and z by rotates the triangle so that
side OA lies along the real axis (you may use polar coordinates).
(c) Let 0, w', and z' be the three vertices of the rotated triangle. Show that
Re[z'] = and Im[2′] = 2
zw+Zw
(d) Show that the area of the rotated triangle is. (Since rotation
doesn't change the area, your formula also gives the area of triangle
OAC.
(e) Let OA'C' denote the rotated triangle. Express the cosine of angle
LA'OC' in terms of w and z.
(f) Let |OA', OC', and A'C' denote the lengths of the three sides of the
rotated triangle. Use complex arithmetic with w and z to prove the law
of cosines:
| A'C'|² = |OA|²+|OC|22|OA||OC| cos(ZA'OC').
(Since rotation does not change lengths or angles, you have also proved
the law of cosines for the original triangle OAC.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2Fe008696a-4f64-4776-8f8c-b091cd859b96%2F6gvy4jq_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 4.4.15. Consider a plane with Cartesian coordinates. Let O be
the point (0,0), let A be the point (a, b), and let C be the point (c,d). Also,
let w = a + bi and z = c+di. We may consider the three complex numbers
0, w, z as representing the vertices of triangle OAC.
(A word to the wise: drawing a picture can be extremely helpful.)
(a) Express the lengths of the three sides of the triangle in terms of u and
z. For example, the length of side OA is |w|.
(b) Show that multiplying 0, w, and z by rotates the triangle so that
side OA lies along the real axis (you may use polar coordinates).
(c) Let 0, w', and z' be the three vertices of the rotated triangle. Show that
Re[z'] = and Im[2′] = 2
zw+Zw
(d) Show that the area of the rotated triangle is. (Since rotation
doesn't change the area, your formula also gives the area of triangle
OAC.
(e) Let OA'C' denote the rotated triangle. Express the cosine of angle
LA'OC' in terms of w and z.
(f) Let |OA', OC', and A'C' denote the lengths of the three sides of the
rotated triangle. Use complex arithmetic with w and z to prove the law
of cosines:
| A'C'|² = |OA|²+|OC|22|OA||OC| cos(ZA'OC').
(Since rotation does not change lengths or angles, you have also proved
the law of cosines for the original triangle OAC.)
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