Step by step
The question
Question: write 4(cos 120° + i sin 120°) in rectangular form.
Rectangular form is a + bi.
4(cos 120° + i sin 120°)
1 Read r and θ
The number in front is the distance: r = 4.
The angle is θ = 120°.
r = 4, θ = 120°
2 Find cos θ and sin θ
From the unit circle: cos 120° = −1/2.
cos 120° = −12
And sin 120° = √3/2.
sin 120° = √32
Careful!
cos gives the real part, across.
sin gives the i part, up.
Do not swap them.
Careful: cos goes with the real part, sin with i
3 Multiply out
Put them in.
= 4(−12 + √32i)
Multiply each part by 4: −2 + 2√3 i.
= −2 + 2√3 i
4 Check
Let's check the distance by Pythagoras: the square root of 4 plus 12 is 4.
That is r.
Right!
r = √4 + 12 = 4
Let's look back.
First: read r and θ.
Then: find cos θ and sin θ.
Then: multiply out.
Last: check.
Answer
= −2 + 2√3 i