Mathematics: Find the product of z1 and z2, where z1 = 7(cos 40° + i sin 40°) and z2 = 6(cos 145° + i sin 145°).

Find the product of z1 and z2, where z1 = 7(cos

Ответ:

miriamnelson7545

31.12.2021

For two complex numbers $z_1=re^{i\theta}=r(\cos\theta+i\sin\theta)$ and $z_2=se^{i\varphi}=s(\cos\varphi+i\sin\varphi)$ , the product is

$z_1z_2=rse^{i(\theta+\varphi)}=rs(\cos(\theta+\varphi)+i\sin(\theta+\varphi))$

That is, you multiply the moduli and add the arguments. You have $z_1=7e^{i40^\circ}$ and $z_2=6e^{i145^\circ}$ , so the product is

$z_1z_2=7\times6(\cos(40^\circ+145^\circ)+i\sin(40^\circ+145^\circ)=42(\cos185^\circ+i\sin185^\circ)=42e^{i185^\circ}$

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Ответ:

xojade

31.12.2021

Mathematics

ANSWER

$\boxed {z_1 z_2 = 42( \cos(185 \degree) + i \sin(185 \degree ))}$

EXPLANATION

The given complex numbers are:

$z_1 = 7( \cos(40 \degree) + i \sin(40 \degree) )$

and

$z_2= 6( \cos(145 \degree) + i \sin(145\degree) )$

Recall that;

If

$z_1 = r_1 ( \cos( \theta_1) + i \sin(\theta_1 ))$

and

$z_2 = r_2 ( \cos(\theta_2) + i \sin(\theta_2 ))$

Then,

$z_1 z_2 = r_1 r_2 ( \cos( \theta_1 +\theta_2) + i \sin(\theta_1 + \theta_2 ))$

This implies that,

$z_1 z_2 = 7 \times 6( \cos( 40 \degree +145 \degree) + i \sin(40 \degree +145 \degree ))$

$z_1 z_2 = 42( \cos(185 \degree) + i \sin(185 \degree ))$

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Ответ:

ccameron1

13.10.2021

Mathematics

64.76% probability that the mean diameter of the sample shafts would differ from the population mean by less than .2 inches

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .