Difference between revisions of "2021 AMC 12B Problems/Problem 13"
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How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta = 0?</cmath><math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math> | How many values of <math>\theta</math> in the interval <math>0<\theta\le 2\pi</math> satisfy<cmath>1-3\sin\theta+5\cos3\theta = 0?</cmath><math>\textbf{(A) }2 \qquad \textbf{(B) }4 \qquad \textbf{(C) }5\qquad \textbf{(D) }6 \qquad \textbf{(E) }8</math> | ||
− | ==Solution== | + | ==Solution 1== |
− | First, move terms to get <math>1+ | + | First, move terms to get <math>1+5\cos 3x=3\sin x</math>. After graphing, we find that there are <math>\boxed{6}</math> solutions (two in each period of <math>5\cos 3x</math>). -dstanz5 |
− | ==Solution | + | ==Solution 2== |
We can graph two functions in this case: <math>5\cos{3x}</math> and <math>3\sin{x} -1 </math>. <cmath>\newline</cmath> | We can graph two functions in this case: <math>5\cos{3x}</math> and <math>3\sin{x} -1 </math>. <cmath>\newline</cmath> | ||
Using transformation of functions, we know that <math>5\cos{3x}</math> is just a cos function with | Using transformation of functions, we know that <math>5\cos{3x}</math> is just a cos function with | ||
− | amplitude 5 and | + | amplitude 5 and period <math>\frac{2\pi}{3}</math>. Similarly, <math>3\sin{x} -1 </math> is just a sin function |
with amplitude 3 and shifted 1 unit downwards. So: | with amplitude 3 and shifted 1 unit downwards. So: | ||
<asy> | <asy> | ||
Line 36: | Line 36: | ||
~ pi_is_3.14 | ~ pi_is_3.14 | ||
+ | |||
+ | ==Video Solution by Hawk Math== | ||
+ | https://www.youtube.com/watch?v=p4iCAZRUESs | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2021|ab=B|num-b=12|num-a=14}} | {{AMC12 box|year=2021|ab=B|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 15:57, 5 March 2021
Contents
Problem
How many values of in the interval satisfy
Solution 1
First, move terms to get . After graphing, we find that there are solutions (two in each period of ). -dstanz5
Solution 2
We can graph two functions in this case: and . Using transformation of functions, we know that is just a cos function with amplitude 5 and period . Similarly, is just a sin function with amplitude 3 and shifted 1 unit downwards. So: We have solutions. ~Jamess2022 (burntTacos)
Video Solution by OmegaLearn (Using Sine and Cosine Graph)
~ pi_is_3.14
Video Solution by Hawk Math
https://www.youtube.com/watch?v=p4iCAZRUESs
See Also
2021 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
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 |
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