In these complex exponential problems, solve for x:
1)e^(i*pi) + 2e^(i*pi/4)=?
2)3+3=3i*sqrt(3)=xe^(i*pi/3)
MY attempt:
I'm not really sure of what they are asking.
For the 1st one I used the e^ix=cos(x)+i*sin(x)
and got -1+sqrt(2) +sqrt(2)i
2) I solved for x and fot (3+3i*sqrt(3))/(1/2+i*sqrt(3)/2)
your 1st answer is correct
#2.
x(1/2 + √3/2 i) = 3+3i√3
x = (3+3√3 i))/[1/2 (1+√3 i)]
now rationalize by multiplying by conjugate
x = 3(1+√3 i)* 2(1-√3 i)/(1-3)
x = -3(1+√3 i))(1-√3 i)
x = -3(1-3)
x = 6
To solve these complex exponential problems, you can use Euler's formula and properties of complex numbers.
1) For the first problem, you are given the expression e^(i*pi) + 2e^(i*pi/4). To simplify this expression, you can use Euler's formula, which states e^(ix) = cos(x) + i*sin(x).
So, e^(i*pi) can be written as cos(pi) + i*sin(pi), which simplifies to -1 + 0i, since cos(pi) = -1 and sin(pi) = 0.
Similarly, e^(i*pi/4) can be written as cos(pi/4) + i*sin(pi/4), which simplifies to sqrt(2)/2 + sqrt(2)/2*i, since cos(pi/4) = sqrt(2)/2 and sin(pi/4) = sqrt(2)/2.
Now, the expression becomes (-1 + 0i) + 2(sqrt(2)/2 + sqrt(2)/2*i). By combining like terms, you get -1 + sqrt(2) + sqrt(2)i.
Therefore, the answer to the first problem is -1 + sqrt(2) + sqrt(2)i.
2) For the second problem, you have the expression 3 + 3i*sqrt(3) = xe^(i*pi/3). Here, you need to find the value of x.
First, let's convert 3 + 3i*sqrt(3) to polar form. The magnitude (r) of the complex number is sqrt((3^2) + (3sqrt(3))^2) = 6.
The argument (theta) of the complex number is tan^(-1)((3sqrt(3))/3) = 60 degrees, which is equivalent to pi/3 radians.
Therefore, 3 + 3i*sqrt(3) can be written as 6 * e^(i*pi/3) in polar form.
To find x, we need to divide both sides of the equation by e^(i*pi/3). This gives x = (3 + 3i*sqrt(3)) / e^(i*pi/3).
Now, we need to convert the denominator to rectangular form using Euler's formula. e^(i*pi/3) can be written as cos(pi/3) + i*sin(pi/3), which simplifies to 1/2 + sqrt(3)/2*i.
Substituting these values into x, we have x = (3 + 3i*sqrt(3)) / (1/2 + sqrt(3)/2*i).
To simplify this complex division, multiply both the numerator and denominator by the complex conjugate of the denominator, which is (1/2 - sqrt(3)/2*i).
By multiplying and simplifying, you should be able to find the value of x.
Note: It is important to be careful with computations involving complex numbers, as errors can easily occur.