Can anyone explain this trigonometric Identity to me?

Question

sin²x + cos²x = 1

I don't get how sin² or cos² can be squared? Can anyone explain it to me?

Answer 1

It's the Pythagorean theorem!

O = opposite
A = adjacent
H = hypotenuse

Pythagorean theorem:
O² + A² = H²

Definitions of sine and cosine:
sin(x) = O / H so O = H * sin(x)
cos(x) = A / H so A = H * cos(x)

Plug the definitions into Pythagorean theorem:
(H sin(x))² + (H cos(x))² = H²
H² sin²(x) + H² cos²(x) = H²
Divide both sides by H²:
sin²(x) + cos²(x) = 1

Answer 2

Draw a circle, radius 1, center on the origin. Pick a point on the circle, say in quadrant 1, (x,y), and draw the radius to that point. Draw the triangle formed by the origin, (x,y), and (x,0). Let the angle at the left be A. sin A is y/1 = y, that's a number, and cos A is x/1 = x. By the Pythagorean Theorem, x² + y² = 1 = sin² Θ + cos² Θ.

Perhaps your question was about notation? sin² x means [ sin(x) ]². x is a number, sin(x) is a function of that number and also a number, and so it can be squared like any other number.

Answer 3

the sin/cos are not squared. the values that sin or cosine represent are squared. --> sin is opposite over hypotenuse. if the opposite side has a length of 4 and the hypotenuse has a length of 5, then sin^2 would be (4/5)^2

^2 means squared.

so, in any triangle, the value of sin^2 plus the value of cos^2 should equal 1. this identity os useful when you do not know a value for the side of a triangle, and you work backwards.

Answer 4

It's just a shorthand notational convention, easier to write than

[sin(x)]^2 + [cos(x)]^2 = 1.