the voltage through a certain wire is ( e = 60sinleft(\frac{pi t}{2}-\frac{pi}{2}\right) ). use an identity…

the voltage through a certain wire is ( e = 60sinleft(\frac{pi t}{2}-\frac{pi}{2}\right) ). use an identity to express this in terms of ( cos\frac{pi t}{2} ).\n( e=square )\n(type an exact answer, using ( pi ) as needed. use integers or fractions for any numbers in the expression.)

the voltage through a certain wire is ( e = 60sinleft(\frac{pi t}{2}-\frac{pi}{2}\right) ). use an identity to express this in terms of ( cos\frac{pi t}{2} ).\n( e=square )\n(type an exact answer, using ( pi ) as needed. use integers or fractions for any numbers in the expression.)

Answer

Explanation:

Step1: Use the co - function identity

The co - function identity for sine and cosine is (\sin(A)=\cos\left(\frac{\pi}{2}-A\right)). In the given expression (E = 60\sin\left(\frac{\pi t}{2}-\frac{\pi}{2}\right)), let (A=\frac{\pi t}{2}-\frac{\pi}{2}).

Step2: Apply the identity

By the co - function identity (\sin\left(\frac{\pi t}{2}-\frac{\pi}{2}\right)=\cos\left[\frac{\pi}{2}-\left(\frac{\pi t}{2}-\frac{\pi}{2}\right)\right]). Simplify the expression inside the cosine function: [ \begin{align*} \frac{\pi}{2}-\left(\frac{\pi t}{2}-\frac{\pi}{2}\right)&=\frac{\pi}{2}-\frac{\pi t}{2}+\frac{\pi}{2}\ &=\pi-\frac{\pi t}{2}\ &=\frac{2\pi - \pi t}{2}\ &=\frac{\pi(2 - t)}{2} \end{align*} ]

Answer:

(E = 60\cos\left(\frac{\pi(2 - t)}{2}\right))