Rearrange formulas to highlight a chosen quantity across the expression types studied.
Isolate a triangle side or principal angle with valid trig conditions
Problem
In a right triangle with \(0~<~\theta~<~\pi/2\) and positive sides, isolate \(\text{opposite}\) in \(\sin(\theta)~=~\text{opposite}/\text{hypotenuse}\). State the conditions and sign result.
Big Picture
What this problem is really about
In the sine ratio, the requested side is already the numerator, so clear the hypotenuse denominator by multiplication. The triangle conditions matter after the rearrangement: an acute angle has positive sine, and a hypotenuse is a positive length. Their product therefore has the positive sign required for the opposite side.
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