BQ#3 – Unit T Concepts 1-3

How do the graphs of sine and cosine relate to each other the others? Emphasize asymptotes in your response.

Sine and cosine are both part of the trig ratio and reciprical identities. All the functions have sin/cos in them so that's how they ultimately relate to each other. When a value in the denominator is equal to zero, we know that is undefined and that is where our asymptote is since asymptotes are undefined. 

Tangent? We know the ratio identitiy is sin/cos. On a Unit Circle, we know that the graph is positive in the first and third quadrant while it is negative in the second and fourth. This leads the the graph to have asymptotes at pi/2 and 3pi/2 since that's where cos=0 and we know anything over 0 is the asymptote. From there, the second quadrant starts with a negative and then in the third quadrant, a positive period. This gives tangent the uphill graph.

Cotangent? As for cotangent, the ratio is opposite of tangent (cos/sin). When sin is 0, we find our asymptotes at 0 and pi on the graph. The period start from being positive in the first quadrant to ending in the negative by the second quadrant on the graph. This gives cotangent its downhill graph. Since both tangent and cotangent only cover up pi units in the x-axis, we see it repeats itself right after an asymptote is drawn.

Secant? Secant is the reciprocal of cosine and has a ratio of 1/cos. So when cos is 0, that's where the asymptote lies. This means it would lie at pi/2 and 3pi/2. In the Unit Circle, we know that secant is only positive in the first and last quadrant. That means in the second and third quadrant, it would be negative. To get from a negative to a positive in an instant, we know that an asymptote will be there when the denominator (cos) is zero. This graph follows the pattern of cosine, however, the differences are the asymptotes.

Cosecant? Cosecant is the reciprical of sine and has a ratio of 1/sin. When sin is equal to zero, we know there will be asymptotes. These would be located at 0 and pi. In the unit circle, we know cosecant is negative in both the first and second quadrant and is positive in both the third and fourth quadrant. This graph follows the pattern as sine, however, the differences are the asymptotes.