Use similar triangles to define, in any right triangle with hypotenuse z, the trigonometric function as cos0= x/z sin0= y/z tan0 = y/x
Hint: The cirle has a radius 1.
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Begin with the unit circle centered at the origin of the cartesian coordinate plane. Draw a ray with endpoint (0,0) that intersects the circle at the point (x,y). From (x,y) drop a perpendicular segment to the x-axis. Connect the origin to the intersection of the perpendicular segment and the x-axis. You will have a right triangle.
For example, if your ray was drawn in the 2nd quadrant, your picture looks like this:
Consider the angle at the origin, call it `theta` ; also name the hypotenuse `z` . (Note that since this is a unit circle, z=1). The side opposite `theta` is the vertical leg, and it has length `|y|` , while the horizontal segment adjacent to `theta` has length `|x|` .
In any right triangle we define the trigonometric ratios as `costheta="adjacent"/"hypotenuse",sintheta="opposite"/"hypotenuse",tantheta="opposite"/"adjacent"`
(We can do this because any two right triangles with the same acute angle `theta` are similar by ``AA~. Thus the ratio of the sides is the same for all such right triangles)
Here `costheta=x/z,sintheta=y/z,tantheta=y/x` . If you change the length of z, say make it kz where k>0 is a real number, then this new triangle is similar to the original and the sides are in the ratio `(kz)/z=k` . Thus the length of the horizontal leg is kx, and the length of the vertical leg is ky. Then `costheta=(kx)/(kz)=x/z;sintheta=(ky)/(kz)=y/z,tantheta=(ky)/(kx)=y/x`
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