**DAY 6 - Tangent and Reciprocal Graphs - Mary H. Bruce - EMAT 6690**

**Objective: **Students will be able to graph y = tan x, y = cot x,
y = csc x and y = sec x

Based on previous discussion we realize that tan x = sin x/cos x. Due
to the rational nature of this function we can guess that the domain will be
restricted. Since cos x is in the denominator we would expect
vertical asymptotes everywhere that cos x = 0. This would correspond with
the x-intercepts of the y = cos x graph. **CLICK****
**here for a GSP animation of the y = tan x graph as developed from the unit
circle. Using Nucalc, we can graph y = cos x and y = tan x simultaneously
to see the relationship between the intercepts and the asymptotes.

One should notice that the period of the tangent graph is half the length of the cosine graph. Since tangent is simply the same as slope (y/x) it only takes an interval of Π for the pattern to repeat itself.

The reciprocal trig functions are as follows: csc x = 1/sin x, sec x = 1/cos x, cot x = 1/tan x. One should deduce that the x-intercepts of the original graphs will now become vertical asymptotes, maximums will now become minimums and vice versa.

Using a GSP animation the inverse relationship
becomes very evident with the opposite motion involved. **CLICK****
**here for an animation of y = sin x and y = csc x. Or a
stationary examination using Nucalc:

**CLICK **here for a GSP animation of y = cos x
and y = sec x.

**CLICK **here for a GSP animation of y = tan x
and y = cot x.

**RETURN to Instructional Unit Outline**