Difference between revisions of "2021 AMC 12B Problems/Problem 13"
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==Solution== | ==Solution== | ||
− | We rearrange to get <cmath>5\cos3\theta = 3\sin\theta - 1.</cmath> | + | We rearrange to get <cmath>5\cos3\theta = 3\sin\theta-1.</cmath> |
We can graph two functions in this case: <math>y=5\cos{3x}</math> and <math>y=3\sin{x} -1 </math>. | We can graph two functions in this case: <math>y=5\cos{3x}</math> and <math>y=3\sin{x} -1 </math>. | ||
Using transformation of functions, we know that <math>5\cos{3x}</math> is just a cosine function with amplitude <math>5</math> and period <math>\frac{2\pi}{3}</math>. Similarly, <math>3\sin{x} -1 </math> is just a sine function with amplitude <math>3</math> and shifted <math>1</math> unit downwards. So: | Using transformation of functions, we know that <math>5\cos{3x}</math> is just a cosine function with amplitude <math>5</math> and period <math>\frac{2\pi}{3}</math>. Similarly, <math>3\sin{x} -1 </math> is just a sine function with amplitude <math>3</math> and shifted <math>1</math> unit downwards. So: |
Revision as of 03:19, 28 January 2022
Contents
[hide]Problem
How many values of in the interval satisfy
Solution
We rearrange to get We can graph two functions in this case: and . Using transformation of functions, we know that is just a cosine function with amplitude and period . Similarly, is just a sine function with amplitude and shifted 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.