why does cos(x)^2+sin(x)^2 always equal one?

Answer 1

If you have a right triangle.
Side 1 = x
Side 2 = y
Hypotenuse = sqrt(x^2 + y^2)

cosine = Side 1/ Hypotenuse; cos^2 = x^2/(x^2 + y^2)
sine = Side 2/Hypotenuse; sin^2 = y^2/(x^2 + y^2)

sin^2 + cos^2 = (x^2 + y^2)/(x^2 + y^2) = 1

Answer 2

This is not an easy question to answer in a venue like this - it really needs a sketch. If you aren't familiar with the terminology I'm going to use, then I apologize in advance.

Draw a circle, centered on the origin with radius 1. This is called the "unit circle". Draw a radius from the origin to some point on the circle in the first quadrant - call the intersection of the radius and the circle point P. You know the length of the radius is 1, as that's the radius of the circle. Make an angle q between the radius and the positive x-axis, measuing anticlockwise. Drop a perpendicular from P to the x-axis, and note that you have formed a right triangle. One leg of the triangle has a length equal to the x co-ordinate of P, the other leg is the y co-ordinate of P and (as we've said) the hypotenuse is 1.

Remember your right-triangle trigonometric definitions - the sine of an angle is the opposite leg over the hypotenuse and the cosine is the adjacent leg over the hypotenuse. Figure out the sine and cosine of q, and you'll find that the x co-ordinate of P is the cosine of q and the y co-ordinate is the sine of q. In the triangle as you've drawn it, the Pythagorean Theorem says that x^2 + y^2 = 1. Since x is the cosine and y the sine, we have that cos^2 q + sin^2 q = 1

If you've followed it this far, then you see that it works for First Quadrant angles. I'll leave it to you to extend the demonstration to angles in the other three quadrants - it works.

This, by the way, is why sin^2 x + cos^2 x = 1 is called the Pythagorean Identity.

Hope this helps.

Answer 3

it's a definition.
First, you draw an "unit" circle, which radius is 1.
Then, choose a point on the circle, call it point A.
Draw a line that perpendicular to the horizontal radius at point B.
AB is called sin( AOB)
OB is called cos (AOB)
Using Pithagore => AB^2 + OB^2 = OA^2
=> sin^2 + cos^2 =1

Answer 4

I don't know, but cos+sin=cossin :)

Answer 5

Because of the pythagorean theorum.

Answer 6

It is a constant.