Question

Let z = equals 38 (cosine (startfraction pi over 8 endfraction) i sine (startfraction pi over 8 endfraction) ) and w = 2 (cosine (startfraction pi over 16 endfraction) i sine (startfraction pi over 16 endfraction) ) . what is the product of zw?

Answer 1

To find the product of complex numbers z and w, we multiply their magnitudes and add their angles.

Step-by-step explanation:

To find the product of zw, we need to multiply the given complex numbers. The product of two complex numbers is calculated by multiplying their magnitudes and adding their angles. Let's calculate:

z = 38 (cos(π/8) + i sin(π/8))

w = 2 (cos(π/16) + i sin(π/16))

Multiplying the magnitudes: |z| * |w| = 38 * 2 = 76

Adding the angles: arg(z) + arg(w) = π/8 + π/16 = 5π/16

So, zw = 76 (cos(5π/16) + i sin(5π/16))

Answer 2

It sounds like you're saying

`z = \frac38 \left(\cos\left(\frac\pi8\right) + i \sin\left(\frac\pi8\right)\right)`

`w = 2 \left(\cos\left(\frac\pi{16}\right) + i \sin\left(\frac\pi{16}\right)\right)`

The product
`zw` is obtained by multiplying the moduli and adding the arguments. In other words

`z = |z| e^(i\arg(z)) \text{ and } w = |w| e^(i\arg(w)) \implies zw = |z||w| e^(i(\arg(z)+\arg(w)))`

where
`e^(it)=\cos(t)+i\sin(t)`, so that

`zw = \frac38*2 \left(\cos\left(\frac\pi8+\frac\pi{16}\right) + i \sin\left(\frac\pi8 + \frac\pi{16}\right)\right) = \boxed{\frac34 \left(\cos\left((3\pi)/(16)\right) + i \sin\left((3\pi)/(16)\right)\right)}`