Question

Express 2 = 10 cos(270) + i sin(270) in rectangular form (a + bi). What is the rectangular form of this expression?

A) 10 - 2i
B) -10 - 2i
C) -10 + 2i
D) 10 + 2i

Answer 1

Rectangular form of this expression is -10 + 2i.

Thus the correct option is (c).

Step-by-step explanation:

The given expression
`\(2 = 10 \cos(270) + i \sin(270)\) is in polar form \(r(\cos(\theta) + i \sin(\theta))\), where \(r = 10\) and \(\theta = 270^\circ`\).

Now, to convert this polar form to rectangular form (a + bi), we use Euler's formula:

`\[a + bi = r \cos(\theta) + i \sin(\theta)\]`

Substitute the values:

`\[a + bi = 10 \cos(270) + i \sin(270)\]`

Evaluate the trigonometric functions:

`\[a + bi = 10 \cdot 0 + i \cdot (-1)\]`

Simplify:

a + bi = -i

Now, compare this result with the given answer choices:

A) 10 - 2i

B)-10 - 2i

C) -10 + 2i

D) 10 + 2i

The correct answer is C) -10 + 2i. This represents the rectangular form of the given polar expression.

Thus the correct option is (c).