Thread: 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.

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 on that domain the sine function is non-negative.

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

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.