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Crazy boy [7]
3 years ago
11

Prove Euler's identity using Euler's formula. e^ix = cos x + i sin x

Mathematics
1 answer:
Korolek [52]3 years ago
3 0

First list all the terms out.

e^ix = 1 + ix/1! + (ix)^2/2! + (ix)^3/3! ...

Then, we can expand them.

e^ix = 1 + ix/1! + i^2x^2/2! + i^3x^3/3!...

Then, we can use the rules of raising i to a power.

e^ix = 1 + ix - x^2/2! - ix^3/3!...

Then, we can sort all the real and imaginary terms.

e^ix = (1 - x^2/2!...) + i(x - x^3/3!...)

We can simplify this.

e^ix = cos x + i sin x

This is Euler's Formula.

What happens if we put in pi?

x = pi

e^i*pi = cos(pi) + i sin(pi)

cos(pi) = -1

i sin(pi) = 0

e^i*pi = -1 OR e^i*pi + 1 = 0

That is Euler's identity.

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There is a bag with only red marbles and blue marbles. The probability of randomly choosing a red marble is 2 9 . There are 45 m
pogonyaev

Answer:

10 red marbles

Step-by-step explanation:

Total= 45 marbles

Probability of red= 2/9

Number of red= 45*2/9= 10

7 0
3 years ago
Derek is going to lay grass in a rectangular space that measures 85 by 32 feet.
adoni [48]

Answer:

2720 feet squared

Step-by-step explanation:

85 * 32 = 2720

3 0
3 years ago
Write the ratio 28 inches to 5 feet as a fraction in simplest form
Schach [20]
5 feet is 5*12=60 inches.  Therefore, our ratio is 28:60. Dividing both terms by 4, we see that the most simple ratio is 7:15.
6 0
3 years ago
Why is the percent increase of 45 to 75 not equal to the percentage crease of 75 to 45 check all that apply.
nevsk [136]

Answer: B and F

Step-by-step explanation:

Since, In the first case original amount = 45

So, the increasing percentage when the original amount increases to 75,

=  \frac{75-45}{45} \times 100 = \frac{30}{45} \times 100= \frac{200}{3} %

In second case original amount = 75

So the decreasing percentage when the original amount decreases to 45,

=    \frac{75-45}{75} \times 100 = \frac{30}{75} \times 100= \frac{40}{1} %

Thus, We found that the ratio of the percent increase is not the same as the percent decrease ( because,   \frac{200}{3}\neq \frac{40}{1}

Also, The original amount of the present increase is different from the original amount for the persons decrease. ( Because, 75\neq 45

Therefore, Option B and F is correct only.



5 0
3 years ago
Read 2 more answers
A bank contains 35 coins, all nickels and quarters. The total value of coins is $8.15. How many of each coin does the bank conta
Lina20 [59]

n + q = 35

n = 35-q

0.05n + 0.25q = 8.15

0.05(35-q) +0.25q = 8.15

1.75 -0.05q + 0.25q =8.15

1.75 + 0.20q = 8.15

0.20q = 6.40

q = 6.40 / 0.20 = 32

32 quarters and 3 nickels

5 0
3 years ago
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