Rewrite 4 sin(x) -1 cos(x) as A sin(x + phi)
A=
phi=
-pi less than phi less than pi
To rewrite the expression 4sin(x) - cos(x) as A sin(x + φ), we need to identify the values of A and φ.
Let's start with A. In the expression A sin(x + φ), A represents the amplitude of the function, which is the maximum value the function reaches. To find A, we can rewrite the given expression as follows:
4sin(x) - cos(x) = A sin(x + φ)
To deal with the coefficients, we'll use the trigonometric identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b). Comparing this identity with our expression, we can see that:
- sin(x) corresponds to sin(a),
- -cos(x) corresponds to cos(b),
- 4 corresponds to cos(a),
- A corresponds to sin(b).
From this comparison, we can conclude that A = 4.
Now let's determine φ.
To find φ, we'll use the trigonometric identity cos(a + b) = cos(a)cos(b) - sin(a)sin(b). Comparing this identity with our expression, we can see that:
- sin(x) corresponds to sin(a),
- -cos(x) corresponds to cos(b),
- 4 corresponds to cos(a),
- A corresponds to sin(b).
From this comparison, we can conclude that φ is the angle whose sine is A (4) and cosine is -1.
We know that the sine is positive in the second and third quadrants, and the cosine is negative in the second and fourth quadrants. Hence, φ can be determined as π/2 (90 degrees) or -π/2 (-90 degrees).
However, we are given that -π < φ < π. Therefore, we can conclude that φ = -π/2 (-90 degrees).
To summarize:
A = 4
φ = -π/2 (-90 degrees)