in the one you helped me with, where did this come from: Where did this part come from?
- (sin^2Ø + cos^2Ø)
you should have come across the identity
sin^2Ø + cos^2Ø = 1 , called the Pythagorean
so I replaced 1 with that expression
oh. I didn't think to connect it with that. Thank you for pointing it out to me.
The expression (sin^2Ø + cos^2Ø) is a fundamental identity in trigonometry known as the Pythagorean identity. It states that for any angle Ø, the square of the sine of Ø plus the square of the cosine of Ø is always equal to 1. This identity plays a crucial role in many trigonometric equations and proofs.
To understand where this identity comes from, you can start with the unit circle. In a unit circle, which has a radius of 1, you can draw a right triangle inside it by taking a point on the circle and dropping a perpendicular line from that point to intersect the x-axis.
Let's call the angle between the positive x-axis and the line connecting the origin and the point on the unit circle Ø. In this triangle, the length of the vertical side is sinØ and the length of the horizontal side is cosØ.
Now, we can use the Pythagorean theorem to find the length of the hypotenuse, which is the distance from the origin to the point on the unit circle. The theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides. Therefore, we have:
(hypotenuse)^2 = (sinØ)^2 + (cosØ)^2
Since the hypotenuse of a unit circle is always equal to 1, we can simplify the equation to:
1 = (sinØ)^2 + (cosØ)^2
And that's where the identity (sin^2Ø + cos^2Ø) = 1 comes from. It shows that for any angle Ø, the sum of the squares of the sine and cosine of Ø is always equal to 1.