Understanding the Pythagorean Identity in Trigonometry: Explained with Examples

cos^2θ+sin^2θ

The expression you provided, cos^2θ + sin^2θ, is an example of the Pythagorean identity in trigonometry

The expression you provided, cos^2θ + sin^2θ, is an example of the Pythagorean identity in trigonometry. According to this identity, for any angle θ, the sum of the squares of the cosine and sine of that angle will always equal 1.

To understand why this is the case, let’s break it down:

First, we need to understand the meaning of cos^2θ and sin^2θ.

cos^2θ represents the square of the cosine of angle θ. This means that you take the value of the cosine function for angle θ and multiply it by itself. Similarly, sin^2θ represents the square of the sine of angle θ, which means multiplying the value of the sine function for angle θ by itself.

Now, let’s look at the two trigonometric functions individually:

The cosine function (cos) gives us the ratio of the adjacent side to the hypotenuse in a right triangle. So, if we have a right triangle with angle θ, the ratio of the length of the adjacent side to the hypotenuse would be cosθ.

The sine function (sin) gives us the ratio of the opposite side to the hypotenuse in a right triangle. So, for the same right triangle, if we take the ratio of the length of the opposite side to the hypotenuse, we get sinθ.

Now, let’s apply the Pythagorean theorem to the right triangle:

The Pythagorean theorem 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 two other sides.

Let’s denote the hypotenuse of the triangle as H, the adjacent side as A, and the opposite side as O.

According to the Pythagorean theorem: H^2 = A^2 + O^2.

Now, if we divide both sides of the equation by H^2, we get:

(H^2 / H^2) = (A^2 / H^2) + (O^2 / H^2).

Since (H^2 / H^2) equals 1, we are left with:

1 = (A^2 / H^2) + (O^2 / H^2).

Now, let’s substitute A with cosθ and O with sinθ:

1 = (cos^2θ / H^2) + (sin^2θ / H^2).

Finally, if we replace H^2 with 1 (since the hypotenuse is always normalized to 1 in trigonometric functions), we get:

1 = cos^2θ + sin^2θ.

This proves the Pythagorean identity: cos^2θ + sin^2θ = 1, which holds true for any angle θ.

So, no matter what the value of θ is, if you square the cosine of θ, add it to the square of the sine of θ, you will always get 1.

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