Inverse Help (1 Viewer)

[chanandlerbong]

Hi guys, I'm doing this exam question and need some assistance.
1. Given y=sininverse(cosx)
(i) Find dy/dx

I don't understand why the answer is
y dash = -1 for 0<x<pi
or = 1 for -pi<x<0

My working out was
y dash = -sinx/absolute values(sinx)
I don't understand the solutions.

And
2. F(x) =[ sin(x-pi/6) + sin(x+pi/6) ] / [ cos(x-pi/6) -cos(x+pi/6) ]
Simplify f(x)
I got this one right, (f(x) = root3)

However, part (ii)
For what values of x is f(x) independent of x. Hence sketch f(x)

Answer for part (ii) is f(x) is independent of x for 0<x<pi

Again I don't understand the inequality.

 

[InteGrand]

Hi guys, I'm doing this exam question and need some assistance.
1. Given y=sininverse(cosx)
(i) Find dy/dx

I don't understand why the answer is
y dash = -1 for 0<x<pi
or = 1 for -pi<x<0

My working out was
y dash = -sinx/absolute values(sinx)
I don't understand the solutions.

And
2. F(x) =[ sin(x-pi/6) + sin(x+pi/6) ] / [ cos(x-pi/6) -cos(x+pi/6) ]
Simplify f(x)
I got this one right, (f(x) = root3)

However, part (ii)
For what values of x is f(x) independent of x. Hence sketch f(x)

Answer for part (ii) is f(x) is independent of x for 0<x<pi

Again I don't understand the inequality.

For the first one, you obtained -sinx/|sin(x)|. This is just equal to the negative of the sign of sin(x). So when sin(x) is positive, -sinx/|sin(x)| = -1, and when sin(x) is negative, -sinx/|sin(x)| = 1. When sin(x) = 0, -sinx/|sin(x)| is undefined.

 

[chanandlerbong]

Yeah, but what I don't understand is the given domain? Like in the solutions it was:

= -1 for 0 less than x less than pi

and = 1 for -pi less than x less than 0

 

[InteGrand]

Yeah, but what I don't understand is the given domain? Like in the solutions it was:

= -1 for 0 less than x less than pi

and = 1 for -pi less than x less than 0

That's because those are the domains where sin(x) is positive and negative, respectively.

 

[chanandlerbong]

Is it necessary to state that in the exam?
Also, how do you find domain and range of this 3sininverse(root(1-x^2))

 

[InteGrand]

Is it necessary to state that in the exam?
Also, how do you find domain and range of this 3sininverse(root(1-x^2))

Yes, you should state that, because the answer isn't fully simplified if left as -sin(x)/|sin(x)|.