Why does sin theta squared plus cos theta squared equal 1?

Answer 1

Sin2(theta) + cos2(theta) = 1 for the same reason that the sides of a right triangle squared equal the hypotenuse squared - The pythagorean theorem.

In the unit circle (origin = (0,0), radius = 1), an angle theta is the angle made by some arbitrary ray drawn from the origin at an angle relative to the x axis. The point of that ray that intersects with the circle is the point (x,y).

Sin(theta) is defined as x, and cos(theta) is defined as y. These are primary trigonometric identities, which link trigonometry with geometry.

Since the points (0,0) (x,0) (x,y) (0,x) describe a right triangle, with (0,x) (0,0) (x,0) being the right angle, then x2 + y2 = 12, or sin2(theta) + cos2(theta) = 1.

If this is not clear, draw a circle around the origin, draw a line from the center to an arbitrary point on the circle, and draw the x and y perpendiculars of that point to each axis. You will see a right triangle. X is sine, Y is cosine, and 1 is hypotenuse. It does not matter if X and/or Y is negative - the squaring will make it positive - and the pythagorean theorem should be visible.