If Vr=[sin(B/2)]*{cos[A+(B-1)r],how do we write Vr in the form,
Vr=f(r) - f(r-1) ?
Thank you!
To write Vr in the form Vr = f(r) - f(r-1), we need to rewrite the expression for Vr and find an expression for f(r) that satisfies the given equation.
Let's start by expanding the expression for Vr:
Vr = [sin(B/2)] * {cos[A + (B-1)r]}
Now, we can rewrite Vr as a subtraction of two terms, f(r) and f(r-1):
Vr = f(r) - f(r-1)
Now, we need to find expressions for f(r) and f(r-1) that will satisfy the equation.
From the given expression of Vr, we can identify that f(r) is the result of multiplying [sin(B/2)] and {cos[A + (B-1)r]}, and f(r-1) is the result of multiplying [sin(B/2)] and {cos[A + (B-1)(r-1)]}:
f(r) = [sin(B/2)] * {cos[A + (B-1)r]}
f(r-1) = [sin(B/2)] * {cos[A + (B-1)(r-1)]}
Therefore, we can substitute these expressions back into the original equation:
Vr = f(r) - f(r-1)
[sin(B/2)] * {cos[A + (B-1)r]} = [sin(B/2)] * {cos[A + (B-1)r]} - [sin(B/2)] * {cos[A + (B-1)(r-1)]}
Now, we can simplify the equation by factoring out [sin(B/2)]:
[sin(B/2)] * {cos[A + (B-1)r]} = [sin(B/2)] * [{cos[A + (B-1)r]} - {cos[A + (B-1)(r-1)]}]
Finally, we can cancel out [sin(B/2)] on both sides of the equation, leaving us with the desired form:
cos[A + (B-1)r] = cos[A + (B-1)r] - cos[A + (B-1)(r-1)]
Therefore, Vr can be written in the form Vr = f(r) - f(r-1) as cos[A + (B-1)r] = cos[A + (B-1)r] - cos[A + (B-1)(r-1)].
I hope this explanation helps you understand how to write Vr in the desired form!