Understanding and Applying the Pythagorean Identity | Explained with Trigonometry and Right Triangles

Pythagorean Identitiessin²a + cos²a =

The missing part of the Pythagorean Identity is equal to 1

The missing part of the Pythagorean Identity is equal to 1. Therefore, the complete Pythagorean Identity is:

sin²a + cos²a = 1

This identity holds true for any angle “a” in a right triangle. It is derived from the Pythagorean Theorem, which states that 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.

In the context of trigonometry, the Pythagorean Identity is a fundamental relationship between the sine and cosine functions. It expresses the fact that the squares of the sine and cosine of an angle always sum up to 1. This identity is useful in many trigonometric proofs and calculations.

To understand how the Pythagorean Identity works, let’s consider a right triangle with an angle “a”. We can define the sine of angle “a” as the ratio of the length of the side opposite to the angle to the length of the hypotenuse. Similarly, the cosine of angle “a” is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

Using these definitions, we can express the trigonometric functions as follows:
sin(a) = opposite / hypotenuse
cos(a) = adjacent / hypotenuse

By squaring both of these equations, we get:
sin²(a) = (opposite / hypotenuse)² = opposite² / hypotenuse²
cos²(a) = (adjacent / hypotenuse)² = adjacent² / hypotenuse²

Adding these two equations together, we have:
sin²(a) + cos²(a) = opposite² / hypotenuse² + adjacent² / hypotenuse²

Since the opposite and adjacent sides are the remaining two sides of the right triangle (not including the hypotenuse), their squares sum up to the square of the hypotenuse according to the Pythagorean Theorem:
opposite² + adjacent² = hypotenuse²

Substituting this into the equation above, we get:
sin²(a) + cos²(a) = hypotenuse² / hypotenuse²

Simplifying, we find:
sin²(a) + cos²(a) = 1

Hence, the Pythagorean Identity states that the sum of the squares of the sine and cosine of any angle in a right triangle is always equal to 1.

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