# When can the length of a line be equal to a circular function?

So, I'm having a bit of trouble trying to grasp this concept. I understand that a circular function like cosine is a ratio of two sides of a triangle in reference to an angle, however, one of my problems is this: Given that line BN is tangent to the Unit Circle at the Y-Axis and the line AT is tangent to the Unit Circle at the X-Axis, and P(coss,sins) is a point in quadrant one, prove that AT = tans

My first instinct was to say this: AT is the line opposite of theta. If O is the origin then tans = AT/OA But that really doesn't help me-- tangent is a function that returns the ration of the opposite side and the adjacent side of a triangle, so how could that ratio ever have the same length of the line it's using -- unless the ratio was one to one? I'm not really sure how I should even think about this problem... any help would be appreciated, thanks!

• You have a ratio of two sides, you say. What if the length of one of the sides was one? What would be the value of the ratio then? – tabstop Feb 2 '14 at 16:36
• in that event it would be one Also in the event the arc length was PI/4 However, I don't know if that satisfies the proof – Nolan Anderson Feb 2 '14 at 16:41
• Well, no it wouldn't be one. If I have a ratio $a/b$, and $b=1$, that does not imply that $a/b=1$.... – tabstop Feb 2 '14 at 16:42
• Ah, sorry, I misread -- And now what you said makes perfect sense! Thanks :D – Nolan Anderson Feb 2 '14 at 16:46

$$\frac{sin(s)}{\cos(s)} = \frac{AT}{1}$$
Then it is easy to see that $AT = \tan(s)$.