The Pythagorean Identity: Exploring The Relationship Between The Sine And Cosine Functions In Trigonometry

sin^2x + cos^2x =

1

The expression sin^2x + cos^2x is a trigonometric identity that holds true for all values of x. This identity is known as the Pythagorean identity and it represents the relationship between the sine and cosine functions in a right triangle.

The Pythagorean identity states that sin^2x + cos^2x = 1. This means that the sum of the squares of the sine and cosine of angle x is always equal to 1.

One way to understand why this identity is true is to consider the unit circle, which is a circle with a radius of 1 centered at the origin of the Cartesian coordinate system. If we draw a line from the origin to a point on the unit circle that makes an angle of x with the positive x-axis, then the length of the horizontal component of this line is cos(x) and the length of the vertical component is sin(x).

Using the Pythagorean theorem, the length of the hypotenuse of this right triangle (which is the radius of the unit circle) is equal to sqrt(cos^2(x) + sin^2(x)). However, since the radius of the unit circle is equal to 1, we have:

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

Squaring both sides of this equation gives:

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

This is the Pythagorean identity, which states that the sum of the squares of the sine and cosine of any angle x is always equal to 1.

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