Question

Can somebody explain how these would be done? The selected answer is incorrect, and I was told "Nice try...express the product by first multiplying the coefficients...then adding your "like term" angles...for instance, cos (2pi/5) + cos (-pi/2) = cos (2pi/5 + -pi/2)...then use the calculator in RADIAN mode to evaluate." Doing those steps, I got the correct constant but a coefficient that was completely off. For the second one, I was told "Good effort...express the quotient by first dividing the coefficients...then subtract your "like term" angles...for instance, cos (2pi/5) - cos (-pi/2) = cos (pi/6 - pi/3)...Finally, use the calculator (in radian MODE) to evaluate."

Answer 1

Solution ( Second Attachment ) : - 2.017 + 0.656i

Solution ( First Attachment ) : 16.140 - 5.244i

Explanation:

Second Attachment : The quotient of the two expressions would be the following,

`6\left[\cos \left((2\pi )/(5)\right)+i\sin \left((2\pi \:)/(5)\right)\right]` ÷
`2√(2)\left[\cos \left((-\pi )/(2)\right)+i\sin \left((-\pi \:)/(2)\right)\right]`

So if we want to determine this expression in standard complex form, we can first convert it into trigonometric form, then apply trivial identities. Either that, or we can straight away apply the following identities and substitute,

( 1 ) cos(x) = sin(π / 2 - x)

( 2 ) sin(x) = cos(π / 2 - x)

If cos(x) = sin(π / 2 - x), then cos(2π / 5) = sin(π / 2 - 2π / 5) = sin(π / 10). Respectively sin(2π / 5) = cos(π / 2 - 2π / 5) = cos(π / 10). Let's simplify sin(π / 10) and cos(π / 10) with two more identities,

( 1 )
`\cos \left((x)/(2)\right)=\sqrt{(1+\cos \left(x\right))/(2)}`

( 2 )
`\sin \left((x)/(2)\right)=\sqrt{(1-\cos \left(x\right))/(2)}`

These two identities makes sin(π / 10) =
`\frac{√(2)\sqrt{3-√(5)}}{4}`, and cos(π / 10) =
`\frac{√(2)\sqrt{5+√(5)}}{4}`.

Therefore cos(2π / 5) =
`\frac{√(2)\sqrt{3-√(5)}}{4}`, and sin(2π / 5) =
`\frac{√(2)\sqrt{5+√(5)}}{4}`. Substitute,

`6\left[ \left\frac{√(2)\sqrt{3-√(5)}}{4}+i\left\frac{√(2)\sqrt{5+√(5)}}{4}\right]` ÷
`2√(2)\left[\cos \left((-\pi )/(2)\right)+i\sin \left((-\pi \:)/(2)\right)\right]`

Remember that cos(- π / 2) = 0, and sin(- π / 2) = - 1. Substituting those values,

`6\left[ \left\frac{√(2)\sqrt{3-√(5)}}{4}+i\left\frac{√(2)\sqrt{5+√(5)}}{4}\right]` ÷
`2√(2)\left[0-i\right]`

And now simplify this expression to receive our answer,

`6\left[ \left\frac{√(2)\sqrt{3-√(5)}}{4}+i\left\frac{√(2)\sqrt{5+√(5)}}{4}\right]` ÷
`2√(2)\left[0-i\right]` =
`-\frac{3\sqrt{5+√(5)}}{4}+\frac{3\sqrt{3-√(5)}}{4}i`,

`-\frac{3\sqrt{5+√(5)}}{4}` =
`-2.01749\dots` and
`\:\frac{3\sqrt{3-√(5)}}{4}` =
`0.65552\dots`

=
`-2.01749+0.65552i`

As you can see our solution is option c. - 2.01749 was rounded to - 2.017, and 0.65552 was rounded to 0.656.

________________________________________

First Attachment : We know from the previous problem that cos(2π / 5) =
`\frac{√(2)\sqrt{3-√(5)}}{4}`, sin(2π / 5) =
`\frac{√(2)\sqrt{5+√(5)}}{4}`, cos(- π / 2) = 0, and sin(- π / 2) = - 1. Substituting we receive a simplified expression,

`6\sqrt{5+√(5)}-6i\sqrt{3-√(5)}`

We know that
`6\sqrt{5+√(5)} = 16.13996\dots` and
`-\:6\sqrt{3-√(5)} = -5.24419\dots` . Therefore,

Solution :
`16.13996 - 5.24419i`

Which rounds to about option b.