The Identity cos^2 θ + sin^2 θ = 1

Understanding the Identity cos² θ + sin² θ = 1

This equation is known as the Pythagorean identity in trigonometry, a fundamental relationship between the sine and cosine of an angle in a right-angle triangle.
This identity can be derived from the definitions of sine and cosine in a right-angled triangle, using the Pythagorean theorem which asserts that, in such triangle, the square of the length of the hypotenuse equals the sum of squares of the other two sides.
This identity holds true for all real numbers θ, it even applies for negative angles and angles greater than 360°.

Applying the Identity

The identity cos² θ + sin² θ = 1 can be rearranged to find either sin θ or cos θ if the other one is known.
For example, if cos θ = 0.6, then sin θ can be found by rearranging the identity as follows: sin² θ = 1 - cos² θ, so sin θ = sqrt(1 - 0.6²).
Similarly, if sin θ = 0.8, then cos θ can be found by rearranging the identity as follows: cos² θ = 1 - sin² θ, so cos θ = sqrt(1 - 0.8²).

Important Points to Remember

All trigonometric identities, including the cos² θ + sin² θ = 1 are crucial when solving trigonometric equations or simplifying expressions.
There are two roots for each of the rearranged Pythagorean identities due to the nature of the square. Each will include a positive and a negative root. This leaves two possible angles for sin θ or cos θ. The correct answer depends on the context of the problem and specific quadrant in which the angle resides.
Both sides of the Pythagorean identities represent real numbers; therefore, care needs to be taken while handling the square roots which include negative numbers.