Advanced Trigonometric Identities: Double Angle, Half Angle, Product-to-Sum

Finding Trigonometric Values from a Triangle:
- Given a scenario where $\sin$ is $-3/5$ , implying the y-coordinate is $-3$ and the hypotenuse is $-5$ (or just $-3$ and $-5$ for y and r in a coordinate plane), the other side (x-coordinate) can be found using the Pythagorean theorem, $x^2 + (-3)^2 = 5^2$ , leading to $x^2 + 9 = 25$ , $x^2 = 16$ , and $x = \pm 4$ . The sign depends on the quadrant.
- In another example, if $x = -7$ and the hypotenuse $r = 25$ , then $(-7)^2 + y^2 = 25^2$ . So, $49 + y^2 = 625$ , $y^2 = 576$ , and $y = \pm \sqrt{576} = \pm 24$ . If $A$ is the angle, then $\sin A = y/r = 24/25$ (adjusting for quadrant if necessary, as speaker mentione