Solving harder calculus questions (1 Viewer)

[pomsky]

Yo!
Completely forgot how to arc-trig lol. Much help needed.
"Solve arcsinx - arccosx= arcsin(3x-2)"
Ans: 0.5,1

fanks

 

[Carrotsticks]

Few ways of doing it. I'll start with what would probably be the most familiar to you.

If I asked you to turn arcsin(x)-arccos(x) into a single arcsin(???) expression, would you be able to do that?

Alternatively, you could add arcsin(x)+arccos(x) to the LHS and balance it by adding pi/2 to the other side since arcsin(x)+arccos(x)=pi/2.

 

[pomsky]

Few ways of doing it. I'll start with what would probably be the most familiar to you.

If I asked you to turn arcsin(x)-arccos(x) into a single arcsin(???) expression, would you be able to do that?

Alternatively, you could add arcsin(x)+arccos(x) to the LHS and balance it by adding pi/2 to the other side since arcsin(x)+arccos(x)=pi/2.

Thanks for replying Carrotsticks!

I forgot the rules/ identities for inverse trig haha.
I ended up solving by taking both sides to sin and then expanding it and solving it from there. How would you turn arcsin(x) - arccos(x) into a single arcsin(f(x)) expression? And why would adding arcsin(x) + arccos(x) to LHS be OK if you balanced it by adding pi/2 on the other side? <>

Finally, what are the common identities of inverse trig?

THANK YOUUU

 

[Carrotsticks]

Thanks for replying Carrotsticks!

I forgot the rules/ identities for inverse trig haha.
I ended up solving by taking both sides to sin and then expanding it and solving it from there. How would you turn arcsin(x) - arccos(x) into a single arcsin(f(x)) expression? And why would adding arcsin(x) + arccos(x) to LHS be OK if you balanced it by adding pi/2 on the other side? <>

Finally, what are the common identities of inverse trig?

THANK YOUUU

1. You pretty much did what I had sort of suggested by taking the sine of both sides.

2. It's because it is an actual identity that arcsin(x)+arccos(x)=pi/2 for all x between -1 and 1 inclusive.

3. Mostly the one I mentioned above. Other than that and some properties of odd functions, that's really just it.

 

[pomsky]

Next Q
How do you find the inverse of f(x) = x/(1+x^2)? And then in terms of x?
Managed to get y = (1-x)/x but only after cancelling a y somehow. The answer looks like the quadratic formula but dunno how to complete the square with x's and y's lol.

Ans: f^-1(x) = (1+ sqrt(1-4x^2)/ 2x)

 

[Carrotsticks]

Swap x and y.

Re-arrange to make it a quadratic in terms of y (so pretend the x is a constant).

Use the quadratic formula to obtain y = *some big expression in terms of x*

You will have a plus/minus to take care of. As the question currently stands, we can't eliminate any branches because the original function was not restricted. As it stands, the inverse is not one-to-one.