Question

Given the complex numbers A₁=6∠30^Oand, (a) Convert to rectangular form; (b) covert to polar and exponential form; (c) calculate A₃=(A₁+A₂), giving your answer in polar form; (d) calculate A₄=A₁A₂, giving your answer in rectangular form; (e) calculate A₅=A₁/A₂, giving your answer in exponential form.

Short answer

(a) Therefore, the rectangular form is A₁= 5. 196+j3.

(b) Therefore, the value A₂ in polar form is 6.403∠ 51.34^Oand in exponential form is 6.403e^J51.34 .

(c) Therefore, the value of theA₃ in polar form is 12.19∠ 41.02^O .

(d) Therefore, the value of A₄in rectangular form is 5.784 +j37.98 .

(e) Therefore, the value of A₅in exponential form is 0.937e^J81.34 .

Step-by-step solution

Define the rectangular form, polar form and exponential form.

In case of the rectangular form a complex number can be represented into the graph by real (horizontal axis) and imaginary axis (vertical axis). For example:, Z = 3 + j8 , here 3 represents the magnitude of the real axis points and 8 represents magnitude of the vertical axis points.

The polar form is represented by the magnitude and angle.

The exponential form is represented by the trigonometric functions like sine and cosine.

Determine the rectangular formof .

The polar number is;

A₁=6∠30^O

Convert the value of A₁into rectangular form

Therefore, the value of A₁in rectangular form is 5. 196+j3. .

Determine the polar form and exponential form of A2 .

The complex number is

A₂= 4+j5

Convert the value of A₂into polar form.

Therefore, the value of A₂in polar form is 6.403∠51.34^O.

Now convert the value of A₂ into exponential form, A∠ $\delta$can be written as Ae^J

into the exponential form.

Therefore the value of A₂ in exponential form is 6.403e^J5134 .

Determine the polar form of A1 and A2 .

Add the values of A₁ and A₂

Now, convert the value of A₃into the polar form.

Therefore the value of A₃ in polar form is 12.19∠ 41.02^o .

Determine the rectangular form of A4.

Multiply the values of A₁ and A₂ .

Therefore, the value of A₄in rectangular form is 5.784 + j37.38 .

Determine the exponential form of A5 .

The A_{2 *}is the complex conjugate ofA₂ .

Therefore

Now, calculate the value of A₅ .

Therefore the value of A₅in exponential form 0.937e^j81.34 .