Question

WHERE DOES THE PYTHAGOREAN IDENTITY SIN2 Θ + COS2 Θ = 1 COME FROM? HOW WOULD YOU USE IT TO FIND THE SINE COSINE AND TANGENT VALUES OF THE ANGLE?

Answer 1

Imagine a trigonometric circle with radius equal to 1.

We can say that the opposite side of the angle generated by the origin is equal to sen Ф and the adjacent equal to cos Ф

With this we can say that:

h² = c² + c²

If r = 1, then h = 1

1² = cos² + sen²

1 = cos² Ф + sen² Ф

You can use this equation a lot of times because it's the fundamental trigonometric relation
, so, when you something like:

sen Ф = cos Ф + 1

you can take from sen² Ф + cos² Ф = 1 that $sen = \sqrt{1 - cos^2}$ and then resolve.

Answer 2

Suppose we have a right- angled triangle with theta has one of the angles ( not 90 degrees) and hypotenuse c , opposite side a and adjacent side b.

then sin α = a/c giving a = c sin α .............(1)

and cos α = b/c giving b = c cos α...............(2)

by Pythagoras theorem:-

a^2 + b^2 = c^2 and from equations (1) and (2):-

a^2 + b^2 = c^2 sin^2 α + c^2 cos^2 α

a^2 + b^2 = c^2 ( sin^2 α + cos^2 α)

Comparing this equation with the Pythagoras equation sin^2 α + cos^2 α must equal 1.