Exploring the Pythagorean Identity: The Relationship Between Sine and Cosine in Trigonometry

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

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

sin^2(x) + cos^2(x) equals 1. This is a fundamental trigonometric identity known as the Pythagorean identity.

To understand why this identity holds true, we need to delve into the definitions of sine and cosine.

In a right-angled triangle, the sine of an angle (denoted as sin) is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle (denoted as cos) is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Using these definitions, let’s examine the Pythagorean theorem. The Pythagorean theorem states that in a right-angled 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.

Applying the Pythagorean theorem to a right-angled triangle, where one angle is x, we have:

sin^2(x) + cos^2(x) = (opposite/hypotenuse)^2 + (adjacent/hypotenuse)^2

Considering a general triangle, we can represent the lengths of the three sides as a, b, and c, where c is the hypotenuse. Then the Pythagorean theorem can be written as:

a^2 + b^2 = c^2

Comparing this equation to the equation for sin^2(x) + cos^2(x), we can substitute a for opposite and b for adjacent. Therefore, the Pythagorean theorem can be rewritten as:

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

Which simplifies to:

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

So, regardless of the value of x, the sum of the squares of sine and cosine will always be equal to 1 according to the Pythagorean identity.

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