Let f(x) = 6.4 sin(x) -6.6 cos(x). What is the maximum and minimum value of this function?
report answers accurate to 4 decimal places
To find the maximum and minimum values of the function f(x) = 6.4 sin(x) - 6.6 cos(x), we can use some trigonometric identities.
First, let's rewrite the function in the form: f(x) = A sin(x) + B cos(x), where A = 6.4 and B = -6.6.
Using the identity sin(x + α) = sin(x)cos(α) + cos(x)sin(α), we can rewrite f(x) as:
f(x) = [A cos(α) - B sin(α)] sin(x) + [A sin(α) + B cos(α)] cos(x)
Comparing this with the standard form of sin(x + α), we can see that:
A cos(α) - B sin(α) = 0
A sin(α) + B cos(α) = 1
Solving these equations simultaneously will give us the values of α (phase angle) and a constant term. Let's solve these equations to find the values of α and the maximum and minimum values of f(x).
From A cos(α) - B sin(α) = 0, we can rearrange the equation to get:
cos(α) = B/A = (-6.6) / (6.4)
To find the value of α, we can use the arccos function:
α = arccos((-6.6) / (6.4))
Evaluating this expression, we get α ≈ 2.8311 radians.
Now, to find the maximum and minimum values of f(x), we can use the identity: sin(x + α) = sin(x)cos(α) + cos(x)sin(α).
sin(x + α) = sin(x)cos(α) + cos(x)sin(α)
sin(x + 2.8311) = sin(x)(-6.6 / 6.4) + cos(x)
Now, let's analyze the function sin(x + 2.8311):
The maximum value of sin(x + 2.8311) is +1 when x = π/2 - 2.8311 + 2πk, where k is any integer.
The minimum value of sin(x + 2.8311) is -1 when x = 3π/2 - 2.8311 + 2πk, where k is any integer.
Substituting these values back into the original function f(x) = 6.4 sin(x) - 6.6 cos(x), we get:
Maximum value of f(x) = 6.4 * 1 - 6.6 * 0 ≈ 6.4
Minimum value of f(x) = 6.4 * (-1) - 6.6 * 0 ≈ -6.4
Therefore, the maximum value of f(x) is approximately 6.4, and the minimum value is approximately -6.4.