Question

How do you write the trigonometric form into a complex number in standard form `1/4(cos225+isin225)`?

Answer 1

`-sqrt2/8-sqrt2/8 i`

Explanation:

Firstly, consider the trig part inside the bracket.

Now `225^@` is an angle in the 3rd quadrant where both the sin and cos ratios are `color(blue)"negative"`

`color(orange)"Reminder"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(cos225^@=-cos(225-180)^@=-cos45^@)color(white)(a/a)|)))`

and `color(red)(|bar(ul(color(white)(a/a)color(black)(sin225^@=-sin(225-180)^@=-sin45^@)color(white)(a/a)|)))`

also `color(red)(|bar(ul(color(white)(a/a)color(black)(sin45^@=cos45^@=1/sqrt2)color(white)(a/a)|)))`

`rArrcos225^@+isin225^@=-1/sqrt2-1/sqrt2 i`

so now back to the original expression.

`1/4(cos225^@+isin225^@)=1/4(-1/sqrt2-1/sqrt2 i)`

distributing gives.

`-1/(4sqrt2)-1/(4sqrt2) i=-sqrt2/8-sqrt2/8 i" in standard form"`