A question about tangent/cotangent (trigonometry)

[zid]

Hi,

Could someone clarify my confusion about the functions tangent/cotangent and their appearance on the unit circle?

I understand perfectly how sin/cos function, and how we draw them in the unit circle. But what I don't understand is how we can say that the function tangent is negative in the second quadrant and positive in the third quadrant. First off, in the definition I have learned, it states that the function tan(x) is the ordinate of the point where the tangent and ray cross. Now, if tangent is indeed the ordinate, how can it be negative in the second quadrant where the ordinate axis is positive? I mean, sin is also the ordinate and it's positive in the second quadrant and that makes perfect sense.

I understand how tan(x) is negative in the second quadrant viewing it from a mathematical point where tan(x)=sin(x)/cos(x), and because sin is positive and cos negative in the second quadrant, tan is also negative, and that makes sense. But looking at the graph it doesn't because tan is right next to sin, and while we say sin is positive we say that tan is negative. I am confused, only explanation I have is that on the unit circle the tan we draw is an absolute value, |tan(x)|. Can someone please help me out? This confuses me a lot.

 

[arthur ohlsten]

Perhaps this will help you
1) draw a unit circle with center at 0,0 on a xy axis
2) at a angle theta,[ 0<theta<90 ] sketch a line from the origin to the circle circumfrence. Drop a perpindicular from the line and circles intersection to the x axis.
3) mark the triangles vertical side sin thea, and the adjacent side cos theta
4) from the point 1,0 sketch a perpindicular line,[ a line tangent to the unit circle]
5) extend the line that is at the angle theta untill it intersects the verticle line ,[the line tangent to the unit circle]
6) mark the vertical lines length x

We now have 2 triangles with the common angle theta
then we can write:
sin theta/ cos theta = x/1
or x= sin theta/cos theta

we shall define x as the tangent theta
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If you do the same construction in quadrant 2 and 3 remember you are using negative values.

Arthur

 

[Loren]

Since the tangent and cotangent functions involve only the ordinate and the abscissa, when one is positive and the other is negative, the ratio of one to the other is negative. When they are both the same, the functions are positive. In QII the abscissa is negative and the ordinate is positive, hence tan and cot are negative. In QIII, the abscissa is negative and the ordinate is negative, hence tan and cot are positive (a negative value divided by a negative value yields a positive value).

 

[zid]

Thanks for the help, but I figured out what was confusing me. When I drew the unit circle I used to draw an additional tangent B(-1,0) on the opposite site, and I measured tan(x) on that new tangent, instead on the one that goes through A(1,0). If I measure tan(x) always on one tangent, everything makes perfect sense including the graphic!

 

[Deleted member 4993]

Hi,

.. in the definition I have learned, it states that the function tan(x) is the ordinate of the point where the tangent and ray cross.

This definition is the source of the term tan(?) - however -

This definition of tan(?) is rarely used in conjunction with unit circle (any more). It is preferable that you remember tan(?) as sin(?)/cos(?) or (opposite)/(base) .