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What is the period of cos2t+sin3t?
- a)pi
- b)2*pi
- c)3*pi
- d)4*pi
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
What is the period of cos2t+sin3t?a)pib)2*pic)3*pid)4*piCorrect answer...
Explanation: Period of cos2t=(2*pi)/2=pi
Period of sin3t=(2*pi)/3
LCM of pi and (2*pi)/3 is 2*pi.
Period of sin3t=(2*pi)/3
LCM of pi and (2*pi)/3 is 2*pi.
Most Upvoted Answer
What is the period of cos2t+sin3t?a)pib)2*pic)3*pid)4*piCorrect answer...
Period of cos2t sin3t
Period of a function is defined as the smallest positive value of T such that f(t+T) = f(t) for all t.
To find the period of cos2t sin3t, we need to find the smallest positive value of T such that cos2(t+T) sin3(t+T) = cos2t sin3t for all t.
Solution:
cos2(t+T) sin3(t+T) = (cos2t cos2T - sin2t sin2T) (sin3t cos3T + cos3t sin3T)
= cos2t sin3t cos2T cos3T + cos2t sin3t sin2T sin3T - sin2t sin3t cos2T sin3T - sin2t sin3t sin2T cos3T
= cos2t sin3t (cos2T cos3T - sin2T sin3T) - sin2t sin3t (cos2T sin3T + sin2T cos3T)
For this expression to be equal to cos2t sin3t, we need:
cos2T cos3T - sin2T sin3T = 1 (1)
cos2T sin3T + sin2T cos3T = 0 (2)
Dividing equation (2) by equation (1), we get:
tan2T = -cot3T
tan2T = -tan(90-3T)
2T = 90-3T + n*180 (where n is an integer)
5T = n*180+90
T = (n*180+90)/5
The smallest positive value of T is when n=1, which gives:
T = (1*180+90)/5 = 54
Therefore, the period of cos2t sin3t is 2*pi/54 = pi/27 or approximately 0.116 radians or approximately 6.67 degrees.
Answer: Option B (2*pi/54)
Period of a function is defined as the smallest positive value of T such that f(t+T) = f(t) for all t.
To find the period of cos2t sin3t, we need to find the smallest positive value of T such that cos2(t+T) sin3(t+T) = cos2t sin3t for all t.
Solution:
cos2(t+T) sin3(t+T) = (cos2t cos2T - sin2t sin2T) (sin3t cos3T + cos3t sin3T)
= cos2t sin3t cos2T cos3T + cos2t sin3t sin2T sin3T - sin2t sin3t cos2T sin3T - sin2t sin3t sin2T cos3T
= cos2t sin3t (cos2T cos3T - sin2T sin3T) - sin2t sin3t (cos2T sin3T + sin2T cos3T)
For this expression to be equal to cos2t sin3t, we need:
cos2T cos3T - sin2T sin3T = 1 (1)
cos2T sin3T + sin2T cos3T = 0 (2)
Dividing equation (2) by equation (1), we get:
tan2T = -cot3T
tan2T = -tan(90-3T)
2T = 90-3T + n*180 (where n is an integer)
5T = n*180+90
T = (n*180+90)/5
The smallest positive value of T is when n=1, which gives:
T = (1*180+90)/5 = 54
Therefore, the period of cos2t sin3t is 2*pi/54 = pi/27 or approximately 0.116 radians or approximately 6.67 degrees.
Answer: Option B (2*pi/54)
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