A.) Once, we draw the sin and cos graphs(sin graph starting and ending at 0 and cos graph starting and ending in its amplitude), we notice that the sin and cos graph are both positive in quadrant one, so the tangent graph is positive because if tan is sin/cos and sin and cos are positive a positive divided by a positive is positive.In quadrant two, the sin graph is positive and the cos graph is negative, so the tan graph is negative because a positive divided by a negative is negative. In quadrant three, the both the sin and cos graph are negative therefore the tan graph is positive because a negative divided by a negative is a positive. In quadrant four, sin is negative and cos is positive, so the tan graph is going to be negative. Tangent has an asymptote whenever cos equals zero. The asymptotes are determined with sin and cos. Asymptotes happen when we get undefined. We get undefined when we divide by zero or when cos is zero because tan is sin/cos.
B.)We know cot is cos/sin by our trig identities. Looking at quadrant one, we noticed both the sin and cos graphs are positive, so the cot graph will be positive. Cotangent is going to have different locations for asymptotes as opposed to tangent.Cotangent is going to have asymptotes whenever sin equals one. In quadrant two, sin is positive, but cos is negative so the cot graph will be negative. In quadrant three, both sin and cos are negative so cotangent is going to be positive. In quadrant four, cos is positive and sin is negative so cotangent will be negative.The asymptotes are determined with sin and cos.Asymptotes happen when we get undefined. We get undefined when we divide by zero or when sin is zero because cot is cos/sin.
C.)Secant is the reciprocal of cosine. In quadrant one, secant is positive and it goes very high up because if you take the reciprocal of a small fraction you will get a really big number. Sec is going to have asymptotes whenever cos equals zero.In quadrant two, cos is negative so sec is going to be negative because it needs to reflect off from the cos graph. In quadrant three, cos is negative, so sec is negative. In quadrant four, cos is positive so sec is positive.
D.)When sin equals zero there is going to be asymptotes. In quadrant one and two, sin is positive so csc is going to be positive or above the x-axis because it reflects off the sin graph.In quadrant three and four, sin is negative so csc will be negative or below the x-axis. The location of the asymptote shapes our graph. We have asymptotes when we get undefined and if we look at csc trig identity it is is 1/sin so sin has to be zero in order for us to get undefined which means asymptotes.