Tangent Function and the Unit CircleDate: 06/06/2002 at 06:43:05 From: Andy Subject: Trigonometric Functions and the Unit Circle Hi Dr. Math, I have a question regarding a response in your archive: Trigonometric Functions and the Unit Circle http://mathforum.org/library/drmath/view/53941.html In this response, it is shown that tan(theta) = infinity when theta = pi/2 Then it is dicussed that as theta passes through pi/2 the picture flips and tan(theta) becomes large negative. What I can't work out is why the picture flips? Why can the tangent line not move to the other side of the circle thus making tan(pi/2+0.01) large positive? Many thanks, Andy Date: 06/06/2002 at 08:29:17 From: Doctor Rick Subject: Re: Trigonometric Functions and the Unit Circle Hi, Andy. Yes, we can draw the vertical tangent line on the left instead of the right (I think this is what you mean). But then the picture flips left-to-right instead of top-to-bottom, and the result turns out the same: P+ |\ | \ | \ | \ | \ | \ | \ | \ opp| \ | \ *********** | **** **** | *** \ ** | ** \ ** | * \ ** |** \ * |* \ * * adj=-1 \ A * +-----------------+-----------------+B * * * * ** * * ** ** ** *** *** **** **** *********** The tangent is the ratio of the y coordinate of P to the x coordinate of P. The y coordinate is now a large *positive* number, and the x coordinate is -1, so y/x is a large *negative* number, just as before. Have I cleared things up for you? - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ Date: 06/06/2002 at 12:06:52 From: Andy Subject: Trigonometric Functions and the Unit Circle Hello, Sorry to keep on about it. If I can refer you to a previous question, Demonstrating Sin, Cos, Tan on the Unit Circle http://mathforum.org/library/drmath/view/54104.html So here tan = FE, which is indeed positive. Now if we flip it to the left, then geometrically, FE is still positive as it is above the horizontal and thus a positive distance. I understand that the ratio is negative and so when pi/2 < theta < pi the geometric representation of tan seems to fall down and this is why I am confused. Many thanks, Andy Date: 06/06/2002 at 14:30:20 From: Doctor Rick Subject: Re: Trigonometric Functions and the Unit Circle Hi, Andy. The tangent function is not just FE, it is FE/OE. (Actually, it would be better to call it EF/OE.) Note what Dr. Jerry said near the bottom of the exchange you cite: tan(t) = side opposite/side adjacent = FE/1 The length OE is 1 in his figure. If you flip the figure left for right, then you must use the *signed* distance OE (positive to the right, negative to the left), just as you use the signed distance for EF (positive going up, negative going down). Thus, as I said, you can either look at the point in the second quadrant where the tangent is a positive number divided by -1, or in the fourth quadrant where the tangent is a negative number divided by 1. Either way it is negative. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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