Understanding the Pythagorean Identity: The Relationship between Sine and Cosine in a Right Triangle

Pythagorean Identity: sin^2(x) + cos^2(x) =

The Pythagorean Identity is a fundamental trigonometric identity that relates the values of sine and cosine of any angle in a right triangle

The Pythagorean Identity is a fundamental trigonometric identity that relates the values of sine and cosine of any angle in a right triangle. It states that the square of sine of an angle, plus the square of cosine of the same angle, is always equal to 1.

The Pythagorean Identity is expressed as: sin^2(x) + cos^2(x) = 1

Let’s break it down and understand why this identity holds true.

Consider a right triangle with one angle measuring x. We can label the sides of the triangle as follows:

– The side opposite angle x is represented by ‘a’.
– The side adjacent to angle x is represented by ‘b’.
– The hypotenuse (the side opposite the right angle) is represented by ‘c’.

Using basic trigonometric ratios, we know that:

– sin(x) = a/c
– cos(x) = b/c

By squaring both sides, we get:

– sin^2(x) = (a/c)^2 = a^2/c^2
– cos^2(x) = (b/c)^2 = b^2/c^2

Now, if we add these two squared trigonometric values together:

sin^2(x) + cos^2(x) = a^2/c^2 + b^2/c^2

Since the fractions have the same denominator (c^2), we can add the numerators:

sin^2(x) + cos^2(x) = (a^2 + b^2)/c^2

In a right triangle, the sum of the squares of the lengths of the legs (a^2 + b^2) is equal to the square of the length of the hypotenuse (c^2). This relationship is known as the Pythagorean Theorem.

Therefore, in our equation:

sin^2(x) + cos^2(x) = (a^2 + b^2)/c^2

becomes:

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

Since any number divided by itself equals 1:

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

And that’s the Pythagorean Identity: sin^2(x) + cos^2(x) = 1.

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