Step by step
The question
Question: write 2(cos 60° + i sin 60°) in rectangular form.
Rectangular form is a + bi.
2(cos 60° + i sin 60°)
1 Read r and θ
The number in front is the distance: r = 2.
The angle is θ = 60°.
r = 2, θ = 60°
2 Find cos θ and sin θ
From the unit circle: cos 60° = 1/2.
cos 60° = 12
And sin 60° = √3/2.
sin 60° = √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.
= 2(12 + √32i)
Multiply each part by 2: 1 + √3 i.
= 1 + √3 i
4 Check
Let's check the distance by Pythagoras: the square root of 1 plus 3 is 2.
That is r.
Right!
r = √1 + 3 = 2
Let's look back.
First: read r and θ.
Then: find cos θ and sin θ.
Then: multiply out.
Last: check.
Answer
= 1 + √3 i