91影视

Use the formula for a circle $(x^2+y^2=r^2)$, and substitute $x=r\cos 胃andy=r\sin 胃.$

The formula for a circle centred at the origin is

$x^2+y^2=r^2$

That is, the distance from the origin to any point $(x,y)$on the circle is the radius $r$of the circle.

Let the angle at the origin be theta $胃$.

Now for the trigonometry.

For an angle $胃$in a right triangle, the trig function $\sin 胃$is the ratio $\frac{oppositeside}{hypotenuse}$. In our case, the length of the side opposite of $胃$is the y-coordinate of our point $(x,y)$, and the hypotenuse is our radius $r$. So:

$\sin 胃=\frac{opp}{hyp}=\frac{y}{r}鈬�y=r\sin 胃$

Similarly, $\cos 胃$is the ratio of the $x$-coordinate in $(x,y)$to the radius $r$:

$\cos 胃=\frac{adj}{hyp}=\frac{x}{r}鈬�x=r\cos 胃$

So we have $x=r\cos 胃andy=r\sin 胃$. Substituting these into the circle formula gives

localid="1651820286878" $$\begin{gathered} x^2+y^2=r^2 \\ {\left(r\cos 胃\right)}^2+{\left(r\sin 胃\right)}^2=r^2 \\ {r}^2{\cos }^2胃+r^2{\sin }^2胃=r^2 \end{gathered}$$

The $r^2$'s all cancel, leaving us with

${\cos }^2胃+{\sin }^2胃=1$

This is often rewritten with the localid="1651820335936" ${\sin }^2胃$term in front, like this:

localid="1651821919173" ${\cos }^2胃+{\sin }^2胃=1$

And that's it. That's really all there is to it. Just as the distance between the origin and any point $\left(x,y\right)$on a circle must be the circle's radius, the sum of the squared values for $\sin 胃$and $\cos 胃$must be 1 for any angle $胃$.