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Find A if 3sin theta +4cos theta =A sin (theta +fie ) a) 7 b) 1 c) 5 d)25 the correct answer is c .Can anyone explain how this is done?
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Find A if 3sin theta +4cos theta =A sin (theta +fie ) a) 7 b) 1 c) 5 d...
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Find A if 3sin theta +4cos theta =A sin (theta +fie ) a) 7 b) 1 c) 5 d...
To find the value of A in the equation 3sin(theta) + 4cos(theta) = A*sin(theta + phi), where phi is a constant angle, we can use the properties of trigonometric functions and algebraic manipulation.

1. Start by rearranging the equation:
3sin(theta) + 4cos(theta) = A*sin(theta + phi)

2. Use the trigonometric identity sin(A + B) = sin(A)*cos(B) + cos(A)*sin(B) to rewrite the right side of the equation:
A*sin(theta + phi) = A*sin(theta)*cos(phi) + A*cos(theta)*sin(phi)

3. Compare the coefficients of sin(theta) and cos(theta) on both sides of the equation:
3 = A*cos(phi)
4 = A*sin(phi)

4. Square both equations to eliminate the trigonometric functions:
3^2 = (A*cos(phi))^2
4^2 = (A*sin(phi))^2

Simplify:
9 = A^2*cos^2(phi)
16 = A^2*sin^2(phi)

5. Use the trigonometric identity sin^2(phi) + cos^2(phi) = 1 to eliminate the trigonometric functions:
9 + 16 = A^2*(sin^2(phi) + cos^2(phi))
25 = A^2

6. Take the square root of both sides:
sqrt(25) = sqrt(A^2)
5 = A

Therefore, the value of A is 5. Thus, the correct answer is option c) 5.
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