Sunday, April 20, 2014

BQ #4: Why is a "normal" tangent graph uphill, but a "normal" cotangent graph downhill?




Let's go back to the unit circle - what appears to be the solid foundation of pre-calculus. 
In the unit circle tangent and cotangent were related by its ratios. Cotangent's ratio was the reciprocal of tangent's ratio. Tangent's ratio is y/x meaning cotangent's ratio would be x/y. Now in this unit we would read that as tangent's ratio being sine/cosine (according to Unit Q's identities) and cotangent's ratio would be cosine/sine
Now what these two graphs have (as well as secant and cosecant) are asymptotes. Remember that this occurs when dividing by 0 leading to undefined. Tangent's asymptotes would occur when cosine equals 0 which would be at pi/2 and 3pi/2. Cotangent's asymptotes would occur when sine equals 0, this would be at 0, pi, and 2pi. (Reflect back on the unit circle, think of these values.) 
Because their asymptotes are placed in different areas, this is going to affect the direction in which they go. Also remember that these graphs WILL NEVER TOUCH the asymptotes, EVER. They will get VERY close but NEVER TOUCH. 
Not only do the asymptotes affect its direction, but look at the different colored areas in the images. Red is the first quadrant of the unit circle, where all trig functions are positive. Green is the second quadrant, where tangent and cotangent will be negative (as well as cosine and secant, but not sine or cosecant.) Orange is the third quadrant, where tangent and cotangent will be positive (the rest of the trig functions will be negative.) Blue is the fourth quadrant, where tangent and cotangent will be negative (as well as sine and cosecant but not cosine or secant.) 

Re-using the images from BQ #3, all thanks again to the wonderful Mrs. Kirch. Her amazing help with these graphs can be found here. 

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