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If a sin theta + b cos theta= A cos(theta+phi).then find A and phi?
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If a sin theta + b cos theta= A cos(theta+phi).then find A and phi?
Solution:
Given equation:
sin(theta) + bcos(theta) = Acos(theta + phi)
Step 1: Expand the right side of the equation
Using the formula cos(A + B) = cosAcosB - sinAsinB, we get:
sin(theta) + bcos(theta) = Acos(theta)cos(phi) - Asin(theta)sin(phi)
Step 2: Equate the coefficients of sin(theta) and cos(theta)
From the given equation, we can equate the coefficients of sin(theta) and cos(theta) on both sides:
1 = -Asin(phi)
b = Acos(phi)
Step 3: Solve for A and phi
Using the equations obtained from Step 2, we can solve for A and phi:
From the first equation, we get: A = -1/sin(phi)
Substitute the value of A in the second equation and solve for phi:
b = (-1/sin(phi))cos(phi)
b = -cos(phi)/sin(phi)
b = -cot(phi)
Therefore, phi = arccot(-b)
Now, substitute the value of phi back into the equation A = -1/sin(phi) to solve for A:
A = -1/sin(arccot(-b))
A = -1/cos(arctan(b))
A = -1/sqrt(1 + b^2)
Therefore, A = -1/sqrt(1 + b^2) and phi = arccot(-b).
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If a sin theta + b cos theta= A cos(theta+phi).then find A and phi?
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If a sin theta + b cos theta= A cos(theta+phi).then find A and phi? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If a sin theta + b cos theta= A cos(theta+phi).then find A and phi? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a sin theta + b cos theta= A cos(theta+phi).then find A and phi?.
If a sin theta + b cos theta= A cos(theta+phi).then find A and phi? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If a sin theta + b cos theta= A cos(theta+phi).then find A and phi? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a sin theta + b cos theta= A cos(theta+phi).then find A and phi?.
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