sin^2 x + cos^2 x
The expression sin^2 x + cos^2 x represents the sum of the squares of the sine and cosine functions for a given value of x
The expression sin^2 x + cos^2 x represents the sum of the squares of the sine and cosine functions for a given value of x.
In trigonometry, the Pythagorean Identity states that for any angle x, sin^2 x + cos^2 x = 1. This identity holds true for all real values of x.
To understand why this is true, let’s first consider a right triangle. In a right triangle, one angle is 90 degrees (π/2 radians), and the other two angles are acute angles. Let’s call one of these acute angles x.
The sine of angle x (sin x) is defined as the ratio of the length of the side opposite the angle to the hypotenuse of the triangle. The cosine of angle x (cos x) is defined as the ratio of the length of the adjacent side to the hypotenuse.
Now, if we take the square of the sine of angle x (sin^2 x), it represents the square of the ratio of the length of the side opposite angle x to the hypotenuse. Similarly, if we take the square of the cosine of angle x (cos^2 x), it represents the square of the ratio of the length of the adjacent side to the hypotenuse.
According to the Pythagorean Theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore, sin^2 x + cos^2 x represents the sum of these two squares, which is equal to 1 (the square of the hypotenuse’s length).
This relationship holds true for all values of x, not just for right triangles. It is a fundamental property of the trigonometric functions, sin and cos.
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