BQ#3 - Unit T Concept 1-3

How do the graph of sine and cosine relate to each other? 

Tangent 

The ratio for tangent is sine/cosine. From this, we can determine how the graph will look based on if sine/cosine is positive or negative in the quadrants I-IV. Therefore that is why the order for tangent is +_+_ on the unit circle. Another thing is that there are asymptotes in tangent which means that there basically where tangent can't exist because of the undefined trig ratio. With the asymptotes in tangent (an example of the asymptotes are pi/2 and 3pi/2) and the order of positive and negative, it results in the shape of the line which is a uphill line. The uphill line comes from quadrant II with the start being in the negative region because of cosine and then going to positive in quadrant III also because of cosine. 

Cotangent 

The ratio for cotangent is cosine/sine. It is essentially like tangent but its relationship is inverse. For starters, the asymptotes are 0 and pi. This is when sine = 0. The order is once again +_+_ but with the asymptotes in place, the graph will change and it will result in a downhill graph because in Quadrant I, cotangent will be positive but moving to Quadrant II, cotangent will become negative and since the asymptotes restrict the graph from continuing forever, our period shows us a downhill graph. 

Secant 

The ratio for secant is r/x. Since secant is the inverse of cosine, if cosine is positive or negative, secant will also be positive or negative. Going on with asymptotes, since secant is 1/cosine, it is similar to tangent which is sine/cosine meaning that secant will have the same asymptotes as tangent which are pi/2 and 3pi/2. Therefore, when drawing it out, the graph will look similar to the sine/cosine graph but the thing to notice is that the secant graph has almost parabola looking graph. 

Cosecant

The Ratio for cosecant is r/y. Since cosecant is the inverse of sine, this means that where sine is positive or negative on the unit circle also goes along with cosecant. The asymptoes for cosecant also go along with cotangent because the denominator for cosecant and cotangent are sine so the asymptotes are 0 and pi. Once again, the graph will look similar to cos/sine except for the asymptotes which cut off the graph and also the parabolas that are within the periods.