sin^2 x + cos^2 x
The expression sin^2 x + cos^2 x is a fundamental identity in trigonometry
The expression sin^2 x + cos^2 x is a fundamental identity in trigonometry. It represents one of the most important relations between the trigonometric functions sine (sin) and cosine (cos). Let’s explore this identity further:
In trigonometry, sine and cosine are defined in terms of a right triangle. Consider a right triangle with an angle x, where the side opposite to x is called the opposite side (denoted as O), the side adjacent to x is called the adjacent side (denoted as A), and the hypotenuse is denoted as H.
Based on this triangle, we can define the trigonometric functions as follows:
sin x = O / H
cos x = A / H
Using the Pythagorean theorem (a^2 + b^2 = c^2), we also have:
O^2 + A^2 = H^2
To express sin^2 x + cos^2 x, we can substitute the definitions of sine and cosine:
(sin x)^2 + (cos x)^2 = (O / H)^2 + (A / H)^2
Now, let’s simplify this expression:
= O^2 / H^2 + A^2 / H^2
Since the numerator and the denominator have a common factor of H^2, we can combine the terms:
= (O^2 + A^2) / H^2
Finally, using the Pythagorean theorem (O^2 + A^2 = H^2), we find:
= H^2 / H^2
This simplifies to:
= 1
Therefore, sin^2 x + cos^2 x = 1.
This result is known as the Pythagorean identity for trigonometric functions. It shows that the sum of the squares of sine and cosine of any angle is always equal to 1. This fundamental identity is widely used in trigonometry and has various applications in mathematics, physics, and engineering.
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