Question

Write the complex number in trigonometric form.
z = 5 - 5i

Answer 1

`\large \boxed{ \boxed{ \red{ \ \rm 5 √(2) ( \cos( (7\pi)/(4) ) + i \sin( (7\pi)/(4) ) )}}}`

Explanation:

to understand this

you need to know about:

• imaginary number
• PEMDAS

tips and formulas:

• 
  `\sf \: formula \: of \: trigonometric \: form : \\ \rm r( \cos( \theta) + i \sin( \theta) )`
• 
  `\rm \: r = \sqrt{ {a}^(2) + {b}^(2) }`
• 
  `\rm \theta = \arctan( (b)/(a) )`

given:

• z=5-5i

let's solve:

complex number:a+bi

therefore

• our a is 5
• our b is -5

`\sf sustitute \: the \: value \: of \: a \: and \: b \: into \: the \: modulus :`

`\rm \: r = \sqrt{ {5}^(2) + ( { - 5)}^(2) }`

`\sf simplify \: squres :`

`\rm \: r = √(25 + 25)`

`\sf simplify \: addition :`

`\rm \: r = √(50)`

`\sf rewrite \: 50 \: as \: 2 * 25`

`\sf r = √(2 * 25)`

`\sf use \: √(ab) \iff \: √(a ) √(b) :`

`\rm r = √(25) √(2)`

`\therefore \rm \: r = 5 √(2)`

`\sf sustitute \: the \: value \: of \: a \: and \: b \: into \: the \: agrument :`

`\rm \theta = \arctan(( - 5)/(5) )`

`\sf simplify \: divition :`

`\rm \theta = \arctan( - 1)`

`\theta = - (\pi)/(4)`

`\sf add \: 2\pi \: to \: get \: a \: positive \: agrument`

`\rm \theta = - (\pi)/(4) + 2\pi \\ \rm\therefore \: \theta = (7\pi)/(4)`

`\sf \: sustitute \: the \: value \: of \: \rm \: r \: and \: \theta : \\ \rm 5 √(2) ( \cos( (7\pi)/(4) ) + i \sin( (7\pi)/(4) ) )`

`\text{And we are done!}`