a When an alternating current of frequency f and peak current I_0 passes through a resistance R, the power delivered to the resistance at time t is given by the following expression.
P=I_0 ^2 R sin^2 2Ï€ ft
We want to rewrite this expression in terms of cos^2 2Ï€ ft. To do so, we will to transform the given expression using one of the Trigonometric Identities. Recall the Pythagorean Identity considering the sum of squares of sine and cosine.
cos^2θ+sin^2θ=1
Let's transform this identity a bit to obtain a formula for sin^2θ.
cos^2θ+sin^2θ=1
⇕
sin^2θ=1-cos^2θ
In our case we have θ=2π ft.
sin^2 2Ï€ ft=1-cos^2 2Ï€ ft
Let's use this formula to transform the given expression.
P=I_0 ^2 R sin^2 2Ï€ ft
⇕
P=I_0 ^2 R (1-cos^2 2Ï€ ft)
We have obtained an expression in terms of cos^2 2Ï€ ft.
b Now we want to rewrite the given expression in terms of csc^2 2Ï€ ft. Recall one of the Reciprocal Identities considering the reciprocal of cosecant.
sinθ=1/cscθ, for cscθ ≠0In our case θ=2π ft.
sin 2π ft=1/csc 2π ft, for csc 2π ft ≠0
⇕
sin^2 2π ft=1/csc^2 2π ft, for csc 2π ft ≠0
Let's now use the above formula to transform the given expression.
