Difference between revisions of "1984 AIME Problems/Problem 8"
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== Problem == | == Problem == | ||
− | The equation <math>z^6+z^3+1</math> has complex | + | The equation <math>z^6+z^3+1</math> has [[complex root]]s with argument <math>\theta</math> between <math>90^\circ</math> and <math>180^\circ</math> in the [[complex plane]]. Determine the degree measure of <math>\theta</math>. |
== Solution == | == Solution == | ||
− | If <math>r</math> is a root of <math>z^6+z^3+1</math>, then <math>0=(r^3-1)(r^6+r^3+1)=r^9-1</math>. The polynomial <math>x^9-1</math> has all of its roots with absolute value 1 and argument of the form <math>40m^\circ</math> for integer <math>m</math>. | + | If <math>r</math> is a root of <math>z^6+z^3+1</math>, then <math>0=(r^3-1)(r^6+r^3+1)=r^9-1</math>. The polynomial <math>x^9-1</math> has all of its roots with [[absolute value]] <math>1</math> and argument of the form <math>40m^\circ</math> for integer <math>m</math>. |
+ | This reduces <math>\theta</math> to either <math>120^{\circ}</math> or <math>160^{\circ}</math>. But <math>\theta</math> can't be <math>120^{\circ}</math> because if <math>r=\cos 120^\circ +i\sin 120^\circ </math>, then <math>r^3=1</math> and <math>r^6+r^3+1=3</math>, a contradiction. This leaves <math>\boxed{\theta=160}</math>. | ||
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== See also == | == See also == | ||
{{AIME box|year=1984|num-b=7|num-a=9}} | {{AIME box|year=1984|num-b=7|num-a=9}} | ||
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+ | [[Category:Intermediate Complex Numbers Problems]] | ||
[[Category:Intermediate Trigonometry Problems]] | [[Category:Intermediate Trigonometry Problems]] | ||
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Revision as of 17:33, 29 October 2007
Problem
The equation has complex roots with argument between and in the complex plane. Determine the degree measure of .
Solution
If is a root of , then . The polynomial has all of its roots with absolute value and argument of the form for integer .
This reduces to either or . But can't be because if , then and , a contradiction. This leaves .
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |