Why does a function of an angle produce a ratio between two sides in sin, cos, and tan? How is that geometrically true?

Answer 1

If I understand your question correctly, the ratios refer to the sides of a right triangle. If c and d are the acute angles of a right triangle, the side opposite c is labeled o and the side adjacent to c is labeled a and the hypotenuse (side opposite the right angle is labeled h, then:

Thus, the trig functions of an acute angle in a right triangle are defined as:

sin c = o/h

cos c = a/h

tan c = o/a

sin d = a/h

cos d = o/h

tan d = a/o

Thus,

sin c = cos d

cos c = sin d

tan c = 1/tan d

I hope that answers your question.