Final answer:
The Pythagorean Identity is validated for any angle due to the extensible relationship between trigonometric functions and the Pythagorean Theorem, which applies to right triangles on a unit circle.
Step-by-step explanation:
The Pythagorean Identity states that for any angle in a unit circle, the sum of the squares of the sine and cosine of that angle is always equal to 1. Specifically, the identity is written as sin^2(\theta) + cos^2(\theta) = 1. This identity can be derived from the Pythagorean Theorem, which relates the sides of a right triangle with lengths a, b, and the hypotenuse c, such that a^2 + b^2 = c^2.
In the context of trigonometry, for any angle \( \theta \), we define the sine and cosine functions based on a right-angled triangle with hypotenuse 1 (the radius of a unit circle), where \( sin(\theta) \) corresponds to a/c and \( cos(\theta) \) to b/c. Because the radius of the unit circle is 1 (which is the hypotenuse of our right triangle), we simply have a as \( sin(\theta) \) and b as \( cos(\theta) \). Substituting these into the Pythagorean Theorem gives us the Pythagorean Identity.
This holds true for any angle, including those greater than 90 degrees, because the definitions of sine and cosine extend to all angles via their respective ratios on the unit circle. This allows for consistent application across all angles.