The Pythagorean Identity: Exploring the Relationship between Sin and Cos in Trigonometry

sin^2x + cos^2x =

The equation sin^2x + cos^2x = 1 is known as the Pythagorean Identity

The equation sin^2x + cos^2x = 1 is known as the Pythagorean Identity. It is a fundamental identity in trigonometry and is used to establish relationships between the trigonometric functions sine and cosine.

To understand how this identity is derived, let’s start with the unit circle. The unit circle is a circle with a radius of 1 centered at the origin (0,0) in a coordinate plane. If we draw a line segment from the origin to a point P(x, y) on the unit circle, we can form a right triangle.

In this right triangle, the angle θ (theta) is the angle between the positive x-axis and the line segment. The coordinates of point P are given by (cos θ, sin θ). The cosine of θ is the x-coordinate (adjacent side) divided by the hypotenuse, which is 1 in this case. Therefore, cos θ = x/1 = x.

Similarly, the sine of θ is the y-coordinate (opposite side) divided by the hypotenuse, which is 1. Therefore, sin θ = y/1 = y.

Using the Pythagorean theorem, we can find that the square of the adjacent side (x^2) plus the square of the opposite side (y^2) is equal to the square of the hypotenuse (1^2 = 1). This can be written as:

x^2 + y^2 = 1

Substituting cos θ for x and sin θ for y, we get:

cos^2θ + sin^2θ = 1

Since we often use x instead of θ, the equation is commonly written as:

cos^2x + sin^2x = 1

This equation holds true for any value of x, as long as x is an angle in radians or degrees.

So, sin^2x + cos^2x = 1 for all values of x.

More Answers: