Tangent: The reason why a normal tangent graph is uphill is because of its unit circle ratio of y/x. Because y/x is the equivalent to sin/cos, we see that whenever cos, or x, equals 0, there will be a vertical asymptote at that point. In this case, a tangent graph has asymptotes at π and 3π/2. Also, tangent is positive in quadrant 1 and 3 while being negative in 2 and 4. We can label the quadrants by every π mark starting at π/2, and label -π/2 as quadrant 4. Since we know in which quadrant tangent is positive, we draw the line ascending from the negative quadrant into the positive quadrant.
Cotangent: The cotangent graph is downhill due to its ratio of x/y. In this case, there is a vertical asymptote whenever sine equals 0. This occurs at π and 2π. The quadrants in which it is negative and positive is relatively the same as tangent. However, the location of the asymptotes starts at π and continues for every subsequent period of π. That means that we have an asymptote starting at the 0 mark and the π mark while it continues infinitely. We label every π/2 interval as a quadrant, and therefore we see that in quadrant 1 the graph descends downward to quadrant 2 where it is positive, constituting its downhill direction.
No comments:
Post a Comment