In 1-2 sentences, describe why there is always a positive and negative solution when solving using the Pythagorean Identity (2 points)

Answer 1

The Pythagorean Identity, which states that \(\sin^2(x) + \cos^2(x) = 1\), allows for both positive and negative solutions for \(\sin(x)\) and \(\cos(x)\) because the unit circle encompasses all angles, resulting in multiple quadrant positions where sine and cosine can take on both positive and negative values while maintaining the equality. As a result, for any given angle, the coordinates of the point on the unit circle can produce both a positive and a negative outcome for these trigonometric functions, reflecting their cyclical nature.