If you don’t put a multiplier ahead of the sin and cos terms, they always fall on a scale of -1 to 1. The equation you wrote is concerned only with the angles. None of those terms relate to radius.

Isnt the 1 supposed to represent the hypotanuse? What if its not 1? Will the equation still work?

Because sin and cos are about ratios, not pure side lengths. For example, 2/1, 4/2, 6/3 are all equal to 2 even though we are dealing with different numbers.

Sin is the same as opposite/hypotenuse and cos is the same as adjacent/hypotenuse.

So the equation is basically:

(opposite/hypotenuse)^(2) + (adjacent/hypotenuse)^(2) = 1

Even if the sides are different, the constraints of a right angled triangle (which is what it is based on, not the unit circle) make all the other sides change such that it still equals 1. Rearranging the equation we get:

opposite^(2)/hypotenuse^(2) + adjacent^(2)/hypotenuse^(2) = 1

opposite^(2) + adjacent^(2) = hypotenuse^(2)

Which is simply the pythagorean theorem.

If you change one side of an equation you have to make exactly the same change to the other side for the equation to remain equal.

One is the default for the functions. If you’re looking for the output to be a length, you need to scale them accordingly. For example, for a circle of radius 3, you’d write:

3 x (sin^2 + cos^2 ) = 3(x1)

The definition of the equation stays the same, but both sides get scaled evenly.

Hope that helps.

Sine is y/r, and cosine is x/r.

If you multiply x, y, and r by the same value - which is what happens if you make the circle bigger - the values of sine and cosine don't change, and therefore neither does the sum of their squares.

If the hypotenuse is not 1, then the x and y sides will not be sine and cosine, but rather a scaled version of them.
Remember that sine and cosine represent ratios, so if the hypotenuse is r, then (x,y)=(rsin(a), rcos(a)), where 'a' is the angle. By pythagorian theorem, x^(2) + y^(2) = r^(2), note that 'r' can be divided on both sides to make the right side 1.

This is because similar triangles have the same ratios. On the unit circle the base length of the triangle is cos(theta) and the height of the triangle is sin(theta) and the hyptenuse has length 1.

If you change the length of the hypotenuse to any other number and call it "r." Then the base and height will change proportionally.

The new triangle will have base length r times cos(theta) and the new height will be r times sin(theta). Now the pythagorean theorem gives you:

(rsin(theta))^2 + (rcos(theta))^2 = r^2

dividing out r^2 leads to the same identity:

sin^2 (theta) + cos^2 (theta) = 1.

In fact if you go back and solve for the base and height of the triangle given by using "r" instead of 1 you end up with the more general forms of sine and cosine:

sin(theta) = y/r

cos(theta) = x/r

where r is the radius of whatever circle you want and (x,y) is the point at which the angle touches the circle of that radius. When you use r=1 you get the original definition of sine and cosine that comes from the unit circle.

This gets into SOH CAH TOA.

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In the unit circle, the length of the hypotenuse is 1. Anything divided by 1 is itself, so we don't have to really think about the division: the sine is just the opposite leg, and the cosine is just the adjacent leg.

But even though we don't have to think about the division in the unit circle, technically it still happens. This is what keeps rhe sine and cosine from changing when the size of the circle does. All circles are geometrically similar, so the sizes of the triangle's legs change in proportion to one another. That means their ratios -the sine and cosine- don't change. The unit circle just makes the math a little easier.

No, the 1 is to represent 1 and nothing else. It comes from the definition of sin and cos in the unit circle.

Sin and Cos are about the angles of the corners of the triangle other than the right angle, they are not about the length of the sides. Sin and Cos are really ratios - given a particular angle the ratio of the sides and the hypotenuse will always be the same (it's a fraction, or a decimal if you will, between 0 and 1). So squaring just the ratios and adding them together will always be 1 because for the ratio itself, the length does't matter.