Tangent
Sine and cosine relate to tangent because when we look back at the Unit Circle, sine and cosine are positive in quadrant one. Based on our knowlwdge of identities, we know that tangent=sine/cosine and since sine and cosine are both positive in the first quadrant, then a positive divided by a positive will result in a positive, which is what tangent is in this quadrant as well. This will cause the tangent graph to be above the x-axis in the first quadrant. In the second quadrant, sine is positive and cosine is negative. If tangent=sin/cos, then this means that tangent will be negative in the second quadrant, since a positive divided by a negative will result in a negative. In the picture, we see that the tangent graph is below the x-axis. In the third quadrant, both sine and cosine are negative, so when you divide two negatives, you get a positive, and tangent is positive as a result of that. In the picture, we see that tangent is once again above the x-axis. In quadrant four, sine is negative and cosine is positive. When divided, this gives you a negative, so tangent will be negative and this will cause it to be below the x-axis. Also, because tangent is sine/cosine, tangent will have asymptotes wherever cosine equals zero on the graph.
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Red=Quadrant 1, Green=Quadrant 2, Orange= Quadrant 3, Blue= Quadrant 4)
Cotangent
In quadrant one, sine and cosine are both positive, so because cotangent=cosine/sine, when this is divided, cotangent will be positive, or above the x-axis. In quadrant two, sine is positive and cosine is negative, so a negative divided by a positive gives you a negative, so cotangent is negative in this quadrant, or below the x-axis. In quadrant three, cotangent will be positive because sine and cosine are bothe negative, so when they are divided, you get a positive. Cotangent will be above the x-axis here. In quadrant four, cosine is positive and sine is negative, so when they are divided, cotangent will result in being negative, or below the x-axis. Also, Cotangent will have asymptotes wherever sine equals zero. Asymptotes only occur when we have undefined, or whenever we divide by zero, so whenever sine=0, there will be an asymptote.
Secant
In quadrant one,cosine is positive, so secant is positive as well. Secant is the reciprocal of cosine. The cosine graph has very small plotted points, so the reciprocal of those numbers is very large. This is why secant goes up so high. The points plotted in the secant graph will be the reciprocals of the cosine graph. Secant will have asymptotes wherever cosine equals zero. In quadrant two, cosine is negative, and so is secant, so secant will be below the x-axis. Therefore, secant also has a point here as well. In quadrant three, cosine is negative, so secant is negative too, so secant will be below the x-axis. In quadrant four, cosine is positive, as well as secant, so secant will be above the x-axis.
Cosecant
In quadrant one and two, sine is positive, so cosecant is positive as well. The cosecant graph will touch the mountain of the sine graph and as it goes higher, it gets closer to the asymptotes. In quadrant three and four, sine is negative, so cosecant will be negative as well, or below the x-axis. The cosecant graph is shaped where the asymptotes are and the asymptotes are where the sine graph is. Cosecant is one over sine, so whenever sine equals zero, cosecant will be divided by zero. Dividing by zero means getting undefined, and undefined means that there will be an asymptote.
