The Pythagorean Identity: Understanding the Relationship Between Sine and Cosine

Pythagorean Identity: sin^2(x) + cos^2(x) =

The Pythagorean Identity is a trigonometric identity that relates the values of sine and cosine for any angle x

The Pythagorean Identity is a trigonometric identity that relates the values of sine and cosine for any angle x. It states that the square of the sine of an angle plus the square of the cosine of the same angle is always equal to 1.

Mathematically, the Pythagorean Identity can be written as:

sin^2(x) + cos^2(x) = 1

This identity holds true for any angle x, whether in degrees or radians. It is one of the fundamental identities in trigonometry and is derived from the Pythagorean Theorem.

To understand this identity further, let’s look at an example. Consider an angle x in a right triangle. Let the length of the opposite side to x be y and the length of the adjacent side be z. According to the Pythagorean Theorem, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides:

y^2 + z^2 = hypotenuse^2

By dividing both sides of this equation by the square of the hypotenuse, we get:

(y^2 / hypotenuse^2) + (z^2 / hypotenuse^2) = 1

Since the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse (y / hypotenuse) and the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse (z / hypotenuse), the equation becomes:

(sin^2(x)) + (cos^2(x)) = 1

Hence, the Pythagorean Identity states that for any angle x, the square of the sine of x plus the square of the cosine of x is always equal to 1.

This identity is not limited to right triangles and can be extended to any angle x on the unit circle or in trigonometric functions. It is widely used in various mathematical applications and serves as a foundation for trigonometric equations and calculations.

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