# Find arcsine(-2) using the rectangular representation of sin w

• richyw
In summary, the student is trying to find the sin of -2 using a textbook equation and is having difficulty understanding what w is. They eventually come to the conclusion that w is the arc sin of -2.
richyw

Find $\sin^{-1}(-2)$ by writing $$sin w = -2[/itex] and using the rectangular representation of $\sin w$ ## Homework Equations Rectangular representation of $\sin w$ ## The Attempt at a Solution I think my biggest problem here is I have literally no idea what the "rectangular representation of $\sin w$ is. My textbook has this formula $sin(x)=\sin(x)\cosh(y)+i\cos(x)\sinh(y)$. Other than that, I am sure where to start here really. I can get the answer using the formula for arcsin, but not from starting with sin... The rectangular representation means z=x+iy for the complex number z. sin-1(w) is a complex number, sin-1(w)=z=x+iy. sin(sin-1(w))=w=-2. The complex Sine function is defined as sin(z)=(eiz-e-iz)/(2i) = sin(x) cosh(y) +i cos(x) sinh(y). So you have the equation sin(x) cosh(y) +i cos(x) sinh(y)=-2. Solve for x, y. ehild thanks. I thought that I needed to use that formula, but I was unaware that it was a definition! I tried doing this but I will try again! nope. I'm still incredibly lost here. I don't even know what w is. That is the definition for sin(z). I am not looking for sin(z). I am looking for sin(w). If sin(w)=sin(x)cosh(y)+icos(x)sinh(y), then I just end up going in circles. my textbook says in order to define arcsin(z), we write w=arcsin(z) when z=sin(w). That's pretty much useless to me. What is w? [tex]w=\sin^{-1}(-2)$$, so sin(w)=-2.

ehild

yes. that was in the initial question.

what is the rectangular representaion of sin(w) though? no matter what I do I just keep ending up back at arcsin(-2), which is where I started.

richyw said:
yes. that was in the initial question.
If it had said, " sin(z) = -2 , find z." could you find z ?

richyw said:
what is the rectangular representaion of sin(w) though? no matter what I do I just keep ending up back at arcsin(-2), which is where I started.

w=x+iy ( x is the real component and y is the imaginary component of the complex number w) .

sin(w)=sin(x) cosh(y) +i cos(x) sinh(y)=-2.

What are x and y? Compare the real parts and the imaginary parts of both sides of the equation. Show your work.ehild

really? that is the definition of w? how does that make it any different from z then?

anyways I am getting stuck even trying to solve that.

I have

sin(x)cosh(y)+i cos(x)sinh(y)=-2

so

sin(x)cosh(y)=-2
cos(x)sinh(y)=0

the second equation says that either cos(x)=0 or sinh(y)=0, if sinh(y)=0 then y=0, but then in the first equation sin(x)=-2 which can't hapen. So then I have

Cos(x)=0
x = π (n - 1/2)

plugging into the first equation is then

$$\sin(\pi(n-1/2))\cosh(y)=-2$$
$$(-1)^n\cosh(y)=-2$$

uh, so I guess$$y=\cosh^{-1}2$$$$x=\pi\left(n-\frac{1}{2}\right)$$

$$y=\ln\left(2+\sqrt{3}\right)$$

Almost correct. Check if x=pi/2 is possible? What has to be the sign of sin(x)? Can cosh(y) negative for real y?

ehild

Last edited:
no. it's not I don't think because I would want cosh(y)=2, so I would need to have sin(x) = -1, so
x = π(2n-1/2) right?

if I do this I think I can hack together something that gives me the correct answer. However I am still uncomfortable with how we can just say w = x+iy . I don't get that at all. To me it seems like saying w = x + iy and then saying sin(w)=z is like saying sin(z)=z

oh wait, I see now why that is not a problem. This all makes sense to me now. Thank you for helping me.

You are welcome z is just an arbitrary complex number. An arbitrary complex number z can be written with it real part (x) and imaginary part (y) as z=x+iy. We have a special complex number now: w=arcsin(-2) which also has a real part and an imaginary part. We could have denoted them something else , u, v instead of x and y to avoid confusionehild

## 1. What is the rectangular representation of sin w?

The rectangular representation of sin w is a way of writing the sine function as a complex number in the form a + bi, where a and b are real numbers and i is the imaginary unit.

## 2. How is arcsine related to the rectangular representation of sin w?

Arcsine, or inverse sine, is the opposite operation of sine and is used to find the angle whose sine value is a given number. In the rectangular representation of sin w, the angle can be found by taking the inverse sine of the imaginary part divided by the real part.

## 3. Why is the value of arcsine sometimes undefined?

The value of arcsine is undefined when the given number is outside the range of -1 to 1. Since the range of sine is -1 to 1, there is no angle whose sine value is -2, making the value of arcsine(-2) undefined.

## 4. How can I find the value of arcsine(-2) using the rectangular representation of sin w?

Using the formula for converting a complex number to polar form, we can find the magnitude and angle of the complex number -2 + 0i. The angle, in this case, will be π/2, which is the value of arcsine(-2) in radians.

## 5. Is there a way to find the value of arcsine(-2) in degrees?

Yes, the value of arcsine(-2) can be converted from radians to degrees by multiplying by 180/π. This will give us the value of 90 degrees, which is the angle corresponding to a complex number with an imaginary part of -2 in the rectangular representation of sin w.

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