# Thread: sin(arcsin(x/3)) why just positive?

1. ## sin(arcsin(x/3)) why just positive?

Which of the following expressions is equal to sin(arccos(x/3))?

The answer is sqrt(9-x^2)/3. I am wondering why it is not "+ or -" sqrt(9-x^2)/3?

My rationale is that cos is positive in both the 1st and 4th quadrant so when I take sin of the angle represented by arccos(x/3) I should have to take the sin from both those quadrants.

2. ## Re: sin(arcsin(x/3)) why just positive?

Originally Posted by Jzon758
Which of the following expressions is equal to sin(arccos(x/3))?

The answer is sqrt(9-x^2)/3. I am wondering why it is not "+ or -" sqrt(9-x^2)/3?

My rationale is that cos is positive in both the 1st and 4th quadrant so when I take sin of the angle represented by arccos(x/3) I should have to take the sin from both those quadrants.
The range of the arccosine of is $\displaystyle [0,\pi]$ on that domain the sine function is non-negative.

3. ## Re: sin(arcsin(x/3)) why just positive?

I see... is there a way to analytically, without a calculator, determine the range of inverse trig functions?

4. ## Re: sin(arcsin(x/3)) why just positive?

Originally Posted by Jzon758
Is there a way to analytically, without a calculator, determine the range of inverse trig functions?
If I were you, I would just learn the domain and range for each.