express u(t)=Tant/1+Tant in F of G
let G = tan(t)
u(t) = g/(1+g)
Let f(g) = g/(1+g)
so, u(t) = f(g)
To express the function u(t) = Tan(t)/(1 + Tan(t)) in terms of F and G, we can use the method of partial fractions.
Step 1: Find the factors of the denominator.
The denominator of u(t) is (1 + Tan(t)). We can rewrite it as (1 + Tan(t)) = (1 + Sin(t))/Cos(t).
Step 2: Write the function using partial fractions.
We can write u(t) as follows:
u(t) = A/Cos(t) + B/Sin(t), where A and B are constants.
Step 3: Find the values of A and B.
To determine the values of A and B, we need to find a common denominator and equate the numerators on both sides.
(1) Multiply both sides by Cos(t)Sin(t) to clear the fractions:
u(t)*Cos(t)*Sin(t) = A*Sin(t) + B*Cos(t).
(2) Substitute t = pi/4 to eliminate one of the terms:
u(pi/4)*Cos(pi/4)*Sin(pi/4) = A*Sin(pi/4) + B*Cos(pi/4).
(3) Simplify the equation:
(u(pi/4)*sqrt(2)/2) = (A + B) / sqrt(2).
Here, we have used the values of Cos(pi/4) and Sin(pi/4), which are 1/sqrt(2) and 1/sqrt(2), respectively.
(4) Solving for A + B, we get:
A + B = (u(pi/4) * sqrt(2))/2.
(5) Substitute t = -pi/4 to eliminate the other term:
u(-pi/4)*Cos(-pi/4)*Sin(-pi/4) = A*Sin(-pi/4) + B*Cos(-pi/4).
(6) Simplify the equation:
(u(-pi/4) * (-sqrt(2))/2) = (-A + B) / sqrt(2).
(7) Solving for -A + B, we get:
-A + B = (u(-pi/4) * sqrt(2))/2.
Step 4: Solve for A and B.
Now, we can solve the system of equations (4) and (7) to find the values of A and B:
A + B = (u(pi/4) * sqrt(2))/2, and
-A + B = (u(-pi/4) * sqrt(2))/2.
Adding these equations, we get:
2B = [(u(pi/4) + u(-pi/4)) * sqrt(2)]/2.
Dividing by 2, we solve for B:
B = [(u(pi/4) + u(-pi/4))/2] * (1/sqrt(2)).
Then, substituting B back into one of the original equations (4 or 7), we can solve for A.
Step 5: Write u(t) in terms of F and G.
Once we have the values of A and B, we can write u(t) in terms of F and G as follows:
u(t) = A/Cos(t) + B/Sin(t)
= [A * 1/(1 - Sin^2(t))] + [B * 1/(Sin(t) * Cos(t))].
Now, substituting the values of A and B, we get:
u(t) = [A * 1/(1 - Sin^2(t))] + [B * 1/(Sin(t) * Cos(t))]
= [(A * 1/(1 - Sin^2(t)))] + [(B * 1/(Sin(t) * Cos(t)))]
= [A * 1/[Cos^2(t)]] + [(B * 1/[Sin(t) * Cos(t)])]
= [A * sec^2(t)] + [(B * [2/(Sin(2t))])].
Therefore, u(t) in terms of F and G is:
u(t) = F(t) + G(t),
where F(t) = A * sec^2(t), and G(t) = B * [2/(Sin(2t))].