1. Find $$ V 1 $$ and $$ V $$ $$ V=10 \cos 2 a $$

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UpStudy AI · Accepted Answer

**Step 1. Write the given function**

We are given the voltage as a function of the angle $$ a $$:
$$
V=10\cos 2a.
$$

**Step 2. Identify $$ V_1 $$**

The notation $$ V_1 $$ suggests that we are to find the first derivative of $$ V $$ with respect to $$ a $$. In other words,
$$
V_1=\frac{dV}{da}.
$$

**Step 3. Differentiate $$ V $$ with respect to $$ a $$**

Differentiate the function $$ V=10\cos2a $$ using the chain rule. Recall that the derivative of $$ \cos u $$ with respect to $$ u $$ is $$-\sin u$$ and then multiply by the derivative of the inner function $$ u $$ with respect to $$ a $$.

Let $$ u=2a $$. Then,
$$
\frac{du}{da}=2.
$$

Now, differentiate:
$$
\frac{dV}{da}=10\frac{d}{da}\bigl(\cos2a\bigr)
=10\bigl[-\sin(2a)\cdot2\bigr]
=-20\sin2a.
$$

Thus, we have:
$$
V_1=-20\sin2a.
$$

**Step 4. Write the final results**

- The original voltage is given by:
  $$
  V=10\cos2a.
  $$
- Its derivative with respect to $$ a $$ is:
  $$
  V_1=-20\sin2a.
  $$
$$ V = 10\cos 2a $$ $$ V_1 = -20\sin 2a $$

1. Find \( V 1 \) and \( V \) \( V=10 \cos 2 a \)

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