I am confused about the Unit Circle explanation for the trigonometric ratios for angles greater than 90 degrees.
It seems that for the first (top right) quadrant, $\sin(\theta)$ is equivalent to the y-coordinate, because
- $\sin(\theta)$ = opposite / hypotenuse
- $\sin(\theta)$ = opposite [hypotenuse is 1]
- $\sin(\theta)$ = y-coordinate [length of opposite side is the y-coordinate]
then, as theta extends in counter-clockwise manner to the top left quadrant, it is assumed that $\sin(\theta)$ is the still the value of the y-coordinate.
From what I understand, the basis for this is that because $\sin(\theta)$ is equivalent to the y-coordinate in the first quadrant, this extends to all quadrants. But this does not make sense to me because the $\sin(\theta)$ = o/h equation was applicable in the first quadrant but not in the others.
It seems to me that there are two definitions for the sine function:
The relationship between the opposite side and the hypotenuse for an acute angle in a right-angled triangle
The y-coordinate of a point along the unit circle, with angle theta (counter-clockwise from the x-axis)
The co-existence of these two definitions is making it confusing for me as it is not clear to me how we can get from the first to the second. How can we have two right angles in a triangle? It is just not possible. How would an "obtuse $sin(\theta)$" look like on the unit circle.
Confused to the core.