Using mathematically precise language, explain in detail how you would multiply the complex number
z1=r1 (cos theta1+isin theta 1)
with the complex number
z2=r2 (cos theta 2+isin theta 2)
To multiply two complex numbers, we use the following formula:
z1 * z2 = r1 * r2 * (cos(theta1 + theta2) + i * sin(theta1 + theta2))
Where:
- z1 = r1 * (cos(theta1) + i * sin(theta1))
- z2 = r2 * (cos(theta2) + i * sin(theta2))
- r1 and r2 are the magnitudes of z1 and z2 respectively
- theta1 and theta2 are the angles of z1 and z2 respectively
By multiplying z1 and z2:
z1 * z2 = r1 * r2 * (cos(theta1) * cos(theta2) - sin(theta1) * sin(theta2) + i * (cos(theta1) * sin(theta2) + sin(theta1) * cos(theta2))
Simplifying this expression, we get:
z1 * z2 = r1 * r2 * (cos(theta1 + theta2) + i * sin(theta1 + theta2))
Therefore, the product of the two complex numbers z1 and z2 is r1 * r2 * (cos(theta1 + theta2) + i * sin(theta1 + theta2).
make it sound like a 12th grader answer this
To multiply two complex numbers, we add their magnitudes and add their angles to get the magnitude and angle of the resulting complex number. In math terms, we use the formula z1 * z2 = r1 * r2 * (cos(theta1 + theta2) + i * sin(theta1 + theta2)) where r1 and r2 are the lengths of the complex numbers, and theta1 and theta2 are the angles they make with the real axis. This formula simplifies to give us the product of the two complex numbers as r1 * r2 * (cos(theta1 + theta2) + i * sin(theta1 + theta2).
show your work not your words
To multiply two complex numbers z1 = r1 (cos theta1 + i sin theta1) and z2 = r2 (cos theta2 + i sin theta2), we can use the formula:
z1 * z2 = r1 * r2 * (cos(theta1 + theta2) + i * sin(theta1 + theta2))
Given z1 = r1 (cos theta1 + i sin theta1) and z2 = r2 (cos theta2 + i sin theta2), we have:
z1 * z2 = r1 * r2 * [cos(theta1) * cos(theta2) - sin(theta1) * sin(theta2) + i * (cos(theta1) * sin(theta2) + sin(theta1) * cos(theta2)]
Using trigonometric identities, we simplify the expression further:
z1 * z2 = r1 * r2 * [cos(theta1 + theta2) + i * sin(theta1 + theta2)]
Therefore, the product of z1 and z2 is r1 * r2 * (cos(theta1 + theta2) + i * sin(theta1 + theta2).