sin^2 x + cos^2 x
The expression sin^2 x + cos^2 x represents a fundamental identity in trigonometry, known as the Pythagorean identity
The expression sin^2 x + cos^2 x represents a fundamental identity in trigonometry, known as the Pythagorean identity. This identity states that for any angle x, the sum of the squares of the sine and cosine of that angle is always equal to 1.
To understand why this is true, let’s start by considering a right triangle. Let one of the acute angles in the triangle be x. The side opposite to x will have length sin x, and the side adjacent to x will have length cos x. The hypotenuse of the triangle will have length 1.
Now, using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can write:
(sin x)^2 + (cos x)^2 = (opposite side)^2 + (adjacent side)^2
= (sin x)^2 + (cos x)^2 = (1)^2
= (sin x)^2 + (cos x)^2 = 1
This equation holds true for any value of x, not just for angles in right triangles. Therefore, we can conclude that sin^2 x + cos^2 x = 1 is a general identity in trigonometry.
This identity is fundamental because it relates the two main trigonometric functions, sine and cosine, showing that they are intimately connected. It is used in various applications of trigonometry and serves as a foundational concept in the subject.
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