I was doing this problem

sin^-1 (4/5) if I'm not mistaking most teachers would say that .927295218 radians is wrong is just an estimation and therefore is wrong so I'm justified in putting this on my paper instead right

( ln( (4i +3)/5)/i )

right? She couldn't mark that wrong... if anythign she should mark .927295218 wrong because it's only an estimation much like using 3.14 for pi

now my question is do I need to show how I got that? Do I need to put the formula... I'm kind of confused by this because if it was sin^-1(1/2) and I just put pi/6 down and didn't show how I got it most teachers wouldn't mark that wrong no...

So am I justified in just putting my answer down or not... if not why... There's no real work to be shown it's just some formula... much like F=ma but in math if I'm not mistaken you don't have to put the formulas like most teachers in math don't require you to put the quadratic equation down on an assignment and then do the work you can show the work if you can't do it in your head but if you can do it in your head most taechers I find with that right...

The only reason why I don't want to show my work is because the teacher gave us boxes to put are work in and the box for that problem is like an inch wide and .5 an inch tall not enough room to put all of my work and I didn't want to write stuff in the margins... plus no real work needs to be shown it's just some simple formula that can be found after doing the work...

If you were my tacher and I just put that down in the box would you take any points off? I don't see how becaues if the problem was sin^-1(1/2) and I just put down pi/6 it would be fine so...

eix= cos x + i sin x

e^ix = 3/5+ 4i/5
ix= 1/5 (3+4i)

x= (3+4i)/5i

Now I don't know exactly what the teacher asked, did she ask for the angle, or not. Few folks would call this an angle. Depending on what I had taught in class, I probably would have counted both correct, but jotted a note on your answer "cute". Actually, I would have specified "answer in radians" on the question.

So on both counts, technially, you are correct: the first is an estimation, the second is "exact" in the DeMor

Oops, sent before I was through. The right hand side should be obviously ln((3+4i)/5i), and DeMor should be de Moivre's complex plane representation.

My instructions were just to "Evaluate" why is that not considered an angle?

I thought that this was already in radians by the way...

ln( (4i +3)/5)/i

why is it not considered to be in radians?

evaluate: Then both are right.

ln((3+4i)/5i) is a representation of an angle, however, it is not commonly used in angle measurement. Again, I would have given it an OK.

When solving trigonometric problems like finding the inverse sine function, it's important to show your work to demonstrate your understanding of the problem and how you arrived at your answer. Here's an explanation of how you can approach solving sin^-1(4/5) and why it's necessary to show your work.

To find the inverse sine of 4/5, you can use either a calculator or a specific formula. Using a calculator, you can directly input the value 4/5 and find its inverse sine, which would be approximately 0.927295218 radians (or about 53.130102°).

However, if you want to manually solve the problem, it's important to show your work by explaining the steps you took. Here's how you can do it:

1. Recall that sin^-1(x) represents the angle whose sine is x.
2. In this case, you need to find the angle whose sine is 4/5.
3. Write sin^-1(4/5) = θ, where θ is the unknown angle.
4. Rewrite this equation as sin(θ) = 4/5.
5. To find the value of θ, take the inverse sine of both sides: θ = sin^-1(4/5).
6. Use a calculator or a trigonometric table to find the value of sin^-1(4/5), which is approximately 0.927295218 radians.

By showing the steps above, you are demonstrating your understanding of the problem and how you arrived at the answer. It also allows your teacher to see your logical reasoning and approach to solving the problem.

Regarding your question about whether you need to show the formula, it's usually not necessary to write down the specific formula since it is typically implied when solving trigonometric problems. However, if your teacher explicitly requires you to show the formula, it's best to follow their instructions.

If you were my student and you only provided the answer without showing any work, it would be difficult for me to assess your understanding of the problem. I would advise you to communicate with your teacher about the space limitation issue for showing your work. It's possible that they may provide additional space or suggest alternative ways to present your work.