How do you prove algebraically whether the following is periodic?
f(x) = 3cos3x + 5sin4x
The sum of two periodic functions is always periodic. In your case, note than when x is a multiple of 2 pi, both functions repeat the values of x = 0.
To prove whether the function f(x) = 3cos(3x) + 5sin(4x) is periodic algebraically, we need to find a value T such that f(x + T) = f(x) for all values of x.
First, let's analyze the periodicity of the individual trigonometric functions within f(x):
- The cosine function, cos(3x), has a period of 2π/3. This means that cos(3x) repeats its values every (2π/3) units of x.
- The sine function, sin(4x), has a period of π/2. This means that sin(4x) repeats its values every (π/2) units of x.
To find a common period for both functions, we need to find the least common multiple (LCM) of their individual periods.
The LCM of 2π/3 and π/2 is (2π/3)(2) = 4π/3.
So, a value of T = 4π/3 can be chosen as a possible period for f(x). Now, we need to prove that f(x + 4π/3) = f(x).
Let's evaluate f(x + 4π/3):
f(x + 4π/3) = 3cos[3(x + 4π/3)] + 5sin[4(x + 4π/3)]
= 3cos(3x + 8π/3) + 5sin(4x + 16π/3)
Now, we can use trigonometric identities to simplify this expression.
cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
Applying these identities, we get:
f(x + 4π/3) = 3[(cos(3x)cos(8π/3)) - (sin(3x)sin(8π/3))] + 5[(sin(4x)cos(16π/3)) + (cos(4x)sin(16π/3))]
Now, let's simplify further using the known values of cos(8π/3) and sin(16π/3):
cos(8π/3) = cos(2π + 2π/3) = cos(2π/3) = -0.5
sin(16π/3) = sin(2π + 10π/3) = sin(10π/3) = sin(2π/3) = sqrt(3)/2
Substituting these values, we get:
f(x + 4π/3) = 3[cos(3x)(-0.5) - sin(3x)(sqrt(3)/2)] + 5[sin(4x)(0.5) + cos(4x)(sqrt(3)/2)]
Expanding and simplifying this expression, we find:
f(x + 4π/3) = -1.5cos(3x) + 2.5sin(4x) - 2.5sin(3x) + 2.5cos(4x)
Now, let's compare this with f(x):
f(x) = 3cos(3x) + 5sin(4x)
By replacing x with x + 4π/3 in f(x), we see that f(x + 4π/3) = f(x).
Therefore, we have proven algebraically that f(x) = 3cos(3x) + 5sin(4x) is periodic with a period of 4π/3.