The Pythagorean Identity [tex]\(\sin^2 \Theta + \cos^2 \Theta = 1\)[/tex] is derived from the Pythagorean Theorem and can be explained through trigonometric relationships in a right triangle. Let's walk through this step-by-step.
1. Understanding the Unit Circle:
Consider a unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane.
2. Identifying Components in a Right Triangle:
Any point on the circumference of the unit circle makes an angle [tex]\(\Theta\)[/tex] with the positive x-axis. You can form a right triangle by drawing a perpendicular from this point to the x-axis.
3. Defining Sine and Cosine:
For an angle [tex]\(\Theta\)[/tex] in this right triangle:
- The length of the adjacent side (adjacent to angle [tex]\(\Theta\)[/tex]) is given by [tex]\( \cos(\Theta) \)[/tex].
- The length of the opposite side (opposite to angle [tex]\(\Theta\)[/tex]) is given by [tex]\( \sin(\Theta) \)[/tex].
- The hypotenuse, which is the radius of the unit circle, has a length of 1.
4. Applying the Pythagorean Theorem:
According to the Pythagorean Theorem, in a right triangle:
[tex]\[
(\text{adjacent side})^2 + (\text{opposite side})^2 = (\text{hypotenuse})^2
\][/tex]
Substituting the values from the unit circle:
[tex]\[
\cos^2(\Theta) + \sin^2(\Theta) = 1^2
\][/tex]
5. Simplifying the Equation:
Since the hypotenuse is 1, the equation simplifies to:
[tex]\[
\cos^2(\Theta) + \sin^2(\Theta) = 1
\][/tex]
This identity is a fundamental result in trigonometry and directly relates to right triangles because it stems from the relationship between the sides of a right triangle inscribed in a unit circle. The lengths corresponding to the cosine and sine of an angle [tex]\(\Theta\)[/tex] satisfy the Pythagorean Theorem, thereby establishing the Pythagorean Identity [tex]\(\sin^2 \Theta + \cos^2 \Theta = 1\)[/tex].
In summary, the Pythagorean Identity [tex]\(\sin^2 \Theta + \cos^2 \Theta = 1\)[/tex] originates from the geometrical properties of right triangles as described by the Pythagorean Theorem, applied to the context of a unit circle.
