sin^2x + cos^2x =
The equation sin^2x + cos^2x = 1 is a fundamental identity in trigonometry
The equation sin^2x + cos^2x = 1 is a fundamental identity in trigonometry. It is known as the Pythagorean identity and holds true for any value of x.
To understand why this equation is true, let’s break it down:
sin^2x represents the sine of x raised to the power of 2.
cos^2x represents the cosine of x raised to the power of 2.
The Pythagorean identity states that the square of the sine of an angle, added to the square of the cosine of the same angle, will always equal 1.
This identity is derived from the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
In trigonometry, the unit circle is used to define the values of sine and cosine for any angle. The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane.
For any point (x, y) on the unit circle, the x-coordinate represents the cosine of the angle formed by the radius from the origin to that point, and the y-coordinate represents the sine of the angle.
Using the unit circle, we can see that for any angle x, sin^2x + cos^2x will always equal 1, since the sum of the squares of the sine and cosine values for any angle on the unit circle is always equal to 1.
Therefore, the equation sin^2x + cos^2x = 1 is a true statement for any value of x.
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