Understanding the Pythagorean Identity | Exploring the Relationship between Trigonometric Functions

Pythagorean identity equal to 1

The Pythagorean identity equal to 1 is:

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

This identity is derived from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides

The Pythagorean identity equal to 1 is:

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

This identity is derived from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In the context of the Pythagorean identity, we consider a unit circle, which is a circle with radius 1 centered at the origin (0, 0) in the Cartesian plane. If we draw an angle x from the positive x-axis in a counterclockwise direction, then the coordinates of the point where the angle intersects the unit circle are (cos(x), sin(x)).

Using this representation, we can see that the x and y-coordinates of a point on the unit circle are given by cos(x) and sin(x) respectively. Now, let’s consider a right triangle formed with one side along the x-axis, one side along the y-axis, and the hypotenuse as the line segment connecting the origin to the point (cos(x), sin(x)).

By applying the Pythagorean theorem to this right triangle, we can see that (cos(x))^2 + (sin(x))^2 = 1^2, which simplifies to cos^2(x) + sin^2(x) = 1. This equation is known as the Pythagorean identity, and it holds true for any angle x.

The Pythagorean identity is a fundamental concept in trigonometry and has numerous applications in mathematical and scientific fields. It allows us to relate the trigonometric functions cosine and sine and has many implications in solving equations, graphing functions, and analyzing periodic phenomena.

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