suppose that at any tiem t(sec) the current i(amp) in an alternating current circuit is i=2cost+2sint. What is the peak (largest magnitude) current for this circuit?
d(i)/dt = -2sint + 2cost
= 0 for a max of i
solving this we get sint = cost
divide by cost
sint/cost = 1
tant = 1
t = pi/4 sec
then i = 2cos(pi/4) + 2sin(pi/4)
= √2 + √2
= 2√2 amps
Well, isn't this current a real joker? It's swinging back and forth with both cosine and sine functions! But fear not, I am here to put a smile on your face and help you out.
To find the peak (largest magnitude) current for this circuit, we need to find the maximum value of the current function i(t). The maximum value occurs when the derivative of the function equals zero.
So, let's calculate the derivative of i(t):
i'(t) = -2sin(t) + 2cos(t)
Now, let's set this derivative equal to zero and solve for t:
-2sin(t) + 2cos(t) = 0
Dividing both sides of the equation by 2, we get:
-sin(t) + cos(t) = 0
Using some trigonometric identities, we can rearrange this equation as:
tan(t) = 1
Ah, the tangent of t is equal to 1! That means t must be one of the special angles where the tangent function equals 1.
One such angle is π/4, which means t = π/4 + 2πn, where n is an integer.
Plugging this value of t back into our current function, we get:
i(π/4 + 2πn) = 2cos(π/4 + 2πn) + 2sin(π/4 + 2πn)
Simplifying this expression, we can find that:
i(π/4 + 2πn) = 2√2 + 2√2 = 4√2
So, the peak current for this circuit is 4√2 amps.
Remember, laughter is the best current... I mean cure!
To find the peak current for the alternating current circuit, we need to find the maximum value of the current function i(t) = 2cos(t) + 2sin(t).
The maximum value of sine and cosine functions is always 1. Therefore, the maximum value of 2sin(t) would be 2, and the maximum value of 2cos(t) would also be 2.
Since both terms have the same maximum value, we can add them together: 2 + 2 = 4.
Hence, the peak current for this circuit is 4 Amperes.
To find the peak (largest magnitude) current in the given alternating current circuit, we need to determine the maximum value of the current function.
The given current function is i = 2cos(t) + 2sin(t).
To find the maximum value, we need to take the derivative of the current function and determine where it equals zero.
Let's start by finding the derivative of i with respect to t:
di/dt = -2sin(t) + 2cos(t)
Now, let's set the derivative equal to zero and solve for t:
-2sin(t) + 2cos(t) = 0
Dividing both sides of the equation by 2, we get:
-sin(t) + cos(t) = 0
Now, let's rewrite sin(t) as cos(t - π/2) using the trigonometric identity:
-cos(t - π/2) + cos(t) = 0
Applying the addition formula for cosine, we have:
cos(t)cos(π/2) + sin(t)sin(π/2) + cos(t) = 0
simplifying:
cos(t) + cos(t) = 0
2cos(t) = 0
cos(t) = 0
Solving for t, we find that t = π/2 or t = 3π/2.
Now, substitute these values back into the original current function to find the peak current:
For t = π/2: i = 2cos(π/2) + 2sin(π/2) = 2(0) + 2(1) = 2
For t = 3π/2: i = 2cos(3π/2) + 2sin(3π/2) = 2(0) + 2(-1) = -2
The larger magnitude current between 2 and -2 is 2.
Therefore, the peak (largest magnitude) current for this circuit is 2 amps.