Question

40 POINTS

Pleaseee help, I need to turn this in soon.

Answer 1

The Pythagorean identity comes from Pythagorean theorem and the unit circle. In the unit circle, for any point on the circle, there y-coordinate represents the sine of the angle and the x-coordinate represents the cosine of the angle. For any angle , we can create a right triangle using the x-coordinate and y-coordinate as legs of the triangle. The hypotenuse of this triangle would be the radius of the unit circle, which is given as 1.

The Pythagorean theorem states that in all right triangles, the following must be true:

, where is the hypotenuse of the triangle and and are two legs of the triangle.

Therefore, we have the following equation:

Answer 2

The Pythagorean identity
`\sin^2\theta+\cos^2\theta=1` comes from Pythagorean theorem and the unit circle. In the unit circle, for any point on the circle, there y-coordinate represents the sine of the angle and the x-coordinate represents the cosine of the angle. For any angle
`\theta`, we can create a right triangle using the x-coordinate and y-coordinate as legs of the triangle. The hypotenuse of this triangle would be the radius of the unit circle, which is given as 1.

The Pythagorean theorem states that in all right triangles, the following must be true:

`a^2+b^2=c^2`, where
`c` is the hypotenuse of the triangle and
`a` and
`b` are two legs of the triangle.

Therefore, we have the following equation:

`(\sin\theta)^2+(\cos\theta)^2=1^2,\\\boxed{\sin^2\theta+\cos^2\theta=1}`