how do I solve this
arcsin(4/5)
note that I am not looking for about 53 degrees
I believe I'm suppose to solve euler's formula for x
i.e.
sin (x) = (e^(ix) - e^(-ix))/(2i)
where x is in radians
hence I would do something like this
sin (x) = (e^(ix) - e^(-ix))/(2i) = 4/5
(e^(ix) - e^(-ix))/(2i) = 4/5
solve the equation above for x
this is were I need help if somebody could just show me quickly how to do this that would be great!!!
If I remeber correctly I need to use cis(x) or something right?
ok this is now an algebra question...
x = 0.92730.. radians
I don't see any way to do this algebraically.
To solve the equation (e^(ix) - e^(-ix))/(2i) = 4/5 and find the value of x, you can use Euler's formula and the concept of complex numbers.
First, let's rewrite the equation using Euler's formula:
(e^(ix) - e^(-ix))/(2i) = 4/5
We know that e^(ix) can be expressed as cos(x) + i*sin(x), and e^(-ix) can be expressed as cos(-x) + i*sin(-x).
So our equation becomes:
((cos(x) + i*sin(x)) - (cos(-x) + i*sin(-x)))/(2i) = 4/5
Simplifying the equation further:
(cos(x) + i*sin(x) - cos(-x) - i*sin(-x))/(2i) = 4/5
Now, we can cancel out the i terms in the numerator:
(cos(x) + i*sin(x) - cos(-x) - i*sin(-x))/(2i) = 4/5
(cos(x) - cos(-x))/(2i) + (i*sin(x) - i*sin(-x))/(2i) = 4/5
(cos(x) - cos(-x))/(2i) is the imaginary part, and (i*sin(x) - i*sin(-x))/(2i) is the real part.
(cos(x) - cos(-x))/(2i) = 0
This implies that cos(x) - cos(-x) = 0, which means cos(x) = cos(-x).
Similarly, (i*sin(x) - i*sin(-x))/(2i) = 4/5
This implies that sin(x) - sin(-x) = (4/5) * 2i
Using the formula for the difference of sine functions:
2*sin(x) = (4/5) * 2i
sin(x) = (4/5) * i
Now, we have sin(x) = (4/5) * i, and we want to find the value of x.
To find the value of arcsin(4/5), we can use the inverse sine function (also known as arcsine).
arcsin(4/5) = x
So, we need to find the angle whose sine is (4/5) * i.
Since the sine function is defined for real numbers, (4/5) * i is not a valid value for the sine function. Therefore, there is no real solution for arcsin(4/5).
In conclusion, the equation (e^(ix) - e^(-ix))/(2i) = 4/5 has no real value solution for x.