Questions: Write the following complex number in the form a+bi. 12(cos 270°+i sin 270°) 12(cos 270°+i sin 270°)=0-12i (Type an exact answer.)

Transcript text: imester Question 19 o Write the following complex number in the form $a+b i$. \[ 12\left(\cos 270^{\circ}+i \sin 270^{\circ}\right) \] $12\left(\cos 270^{\circ}+i \boldsymbol{\operatorname { s i n }} 270^{\circ}\right)=0-12 i$ (Type an exact answer.)

Solution

Solution Steps

Step 1: Identify the trigonometric values

Evaluate $ \cos 270^{\circ} $ and $ \sin 270^{\circ} $: \[ \cos 270^{\circ} = 0, \quad \sin 270^{\circ} = -1. \]

Step 2: Substitute the values into the expression

Substitute $ \cos 270^{\circ} = 0 $ and $ \sin 270^{\circ} = -1 $ into the given complex number: \[ 12\left(\cos 270^{\circ} + i \sin 270^{\circ}\right) = 12(0 + i(-1)). \]

Step 3: Simplify the expression

Simplify the expression: \[ 12(0 + i(-1)) = 0 - 12i. \]