Difference between revisions of "2019 AMC 12B Problems/Problem 17"

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==Solution==
 
==Solution==
  
Convert z and z^3 into <cmath>r\text{cis}\theta</cmath> form, giving <cmath>z=r\text{cis}\theta</cmath> and <cmath>z^3=r^3\text{cis}(3\theta)</cmath>. Since the distance from 0 to z is r, the distance from 0 to z^3 must also be r, so r=1. Now we must find <cmath>\text{cis}(2\theta)=60</cmath>. From 0 < theta < pi/2, we have <cmath>\theta=\frac{\pi}{2}</cmath> and from pi/2 < theta < pi, we see a monotonic decrease of <cmath>\text{cis}(2\theta)</cmath>, from 180 to 0. Hence, there are 2 values that work for 0 < theta < pi. But since the interval pi < theta < 2pi is identical, because 3theta=theta, we have 4 solutions. There are not infinitely many solutions since the same four solutions are duplicated. (D)
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Convert z and z^3 into <cmath>r\text{cis}\theta</cmath> form, giving <cmath>z=r\text{cis}\theta</cmath> and <cmath>z^3=r^3\text{cis}(3\theta)</cmath>. Since the distance from 0 to z is r, the distance from 0 to z^3 must also be r, so r=1. Now we must find <cmath>\text{cis}(2\theta)=60</cmath>. From 0 < theta < pi/2, we have <cmath>\theta=\frac{\pi}{2}</cmath> and from pi/2 < theta < pi, we see a monotonic decrease of <cmath>\text{cis}(2\theta)</cmath>, from 180 to 0. Hence, there are 2 values that work for 0 < theta < pi. But since the interval pi < theta < 2pi is identical, because 3theta=theta at pi, we have 4 solutions. There are not infinitely many solutions since the same four solutions are duplicated. (D)
  
 
-FlatSquare
 
-FlatSquare

Revision as of 15:37, 14 February 2019

Problem

Solution

Convert z and z^3 into \[r\text{cis}\theta\] form, giving \[z=r\text{cis}\theta\] and \[z^3=r^3\text{cis}(3\theta)\]. Since the distance from 0 to z is r, the distance from 0 to z^3 must also be r, so r=1. Now we must find \[\text{cis}(2\theta)=60\]. From 0 < theta < pi/2, we have \[\theta=\frac{\pi}{2}\] and from pi/2 < theta < pi, we see a monotonic decrease of \[\text{cis}(2\theta)\], from 180 to 0. Hence, there are 2 values that work for 0 < theta < pi. But since the interval pi < theta < 2pi is identical, because 3theta=theta at pi, we have 4 solutions. There are not infinitely many solutions since the same four solutions are duplicated. (D)

-FlatSquare

Someone pls help with LaTeX formatting, thanks

See Also

2019 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions