Step by step
The question
Question: write 6(cos 330° + i sin 330°) in rectangular form.
Rectangular form is a + bi.
6(cos 330° + i sin 330°)
1 Read r and θ
The number in front is the distance: r = 6.
The angle is θ = 330°.
r = 6, θ = 330°
2 Find cos θ and sin θ
From the unit circle: cos 330° = √3/2.
cos 330° = √32
And sin 330° = −1/2.
sin 330° = −12
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.
= 6(√32 + (−12)i)
Multiply each part by 6: 3√3 − 3i.
= 3√3 − 3i
4 Check
Let's check the distance by Pythagoras: the square root of 27 plus 9 is 6.
That is r.
Right!
r = √27 + 9 = 6
Let's look back.
First: read r and θ.
Then: find cos θ and sin θ.
Then: multiply out.
Last: check.
Answer
= 3√3 − 3i