Question

Specila function values fing the value of cos(3-i)

Answer 1

The value of
`\(\cos(3 - i)\)` is
`\((3√(10))/(10)\)` after expressing 3 - i in polar form using Euler's formula and rationalizing the denominator.

To find the cosine of a complex number, you can use Euler's formula:
`\(e^(ix) = \cos(x) + i\sin(x)\)`.

Let's express 3 - i in polar form:

`\[ r = √(3^2 + (-1)^2) = √(10) \]\\ \theta = \arctan\left((-1)/(3)\right) \]`

Now, substitute these values into Euler's formula:

`\[ 3 - i = √(10) \cdot \cos(\theta) + i√(10) \cdot \sin(\theta) \]`

Since
`\(\cos(\theta) = (3)/(√(10))\) and \(\sin(\theta) = -(1)/(√(10))\)`, we get:

`\[ 3 - i = √(10) \left((3)/(√(10)) + i\left(-(1)/(√(10))\right)\right) \]`

Now, extract the real part:

`\[ \cos(3 - i) = (3)/(√(10)) \]`

Rationalize the denominator:

`\[ \cos(3 - i) = (3√(10))/(10) \]`