Rewrite 4sin(x)-5cos(x) as Asin(x+phi)

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To rewrite the expression 4sin(x) - 5cos(x) in the form Asin(x + phi), where A and phi are constants, we can make use of trigonometric identities.

First, we know that sin(x + phi) = sin(x)cos(phi) + cos(x)sin(phi). By comparing this identity to the given expression, we can find suitable values for A and phi.

Let's focus on the coefficients of sin(x) and cos(x) separately:

For sin(x): The coefficient is 4. Comparing this to sin(x)'s coefficient in the identity, we can set A*cos(phi) = 4. This means A is the magnitude of the coefficient, which is 4, and cos(phi) is the ratio of the two coefficients, which is 4/A = 4/4 = 1. Therefore, A = 4 and cos(phi) = 1.

For cos(x): The coefficient is -5. Comparing this to cos(x)'s coefficient in the identity, we can set sin(phi) = -5. Since sin^2(phi) + cos^2(phi) = 1, we can solve for cos(phi) using the Pythagorean identity. Since sin(phi) = -5, we have (-5)^2 + cos^2(phi) = 1, which gives cos^2(phi) = 1 - 25 = -24. Since cos(phi) must be a real number, we can't have a negative value under the square root. Therefore, there is no real solution for cos(phi).

Therefore, it is not possible to rewrite the expression 4sin(x) - 5cos(x) in the form Asin(x + phi) where A and phi are real constants.

4sin(x)-5cos(x)

= √41 (4/√41 sinx - 5/√41 cosx)
so,
cosØ = 4/√41
sinØ = -5/√41
Ø = -0.896
Note that Ø is in QIV, since sinØ is negative