Graphing the Other Trigonometric FunctionsDR. SHILDNECKFALL

Graphing the Secant and Cosecant Functions

Trigonometric Relationships What is the relationship between the sine and cosecant functions, and the cosine and secant functions?

They are RECIPROCALS

Trigonometric Relationships What are the important implications of these relationships?

1. Whenever the sine/cosine = 0, the cosecant/secant is ___________

2. Whenever the sine/cosine = 1, the cosecant/secant = _______

3. Whenever the value of the sine/cosine is small (close to zero) the value of the cosecant/secant will be ___________

Undefined1

Large

Trigonometric Relationships What are the important implications of these relationships?

Since the value of the cosecant/secant is undefined when the sine/cosine is zero, the graph of the cosecant/secant will have ________________ at those x-values.

Since the value of the cosecant/secant is one when the value of the sine/cosine is one, the graphs have (only) those points in common.

asymptotes

Graphing the Reciprocals of Sinusoids

Since the secant and cosecant functions are the reciprocals of the cosine and sine functions, we can use those functions to guide our graphs.

First, lightly sketch the graph of the sine (for cosecant) or cosine (for secant) utilizing all of the same transformations that can by picked out from the function.

Second, sketch asymptotes for each of the “zeros” of the sinusoidal function (since the reciprocal of zero is undefined).

Finally sketch a “U” between each asymptote using the max/min of the sinusoid as the vertex. Each “U” should come in along the asymptote, level off, pass through the max/min point, then go out along the other asymptote.

Graph:

Graph:

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Graphing the Tangent and Cotangent Functions

Characteristics of Tangent Functions

While a tangent function is periodic (repeats itself) its graph has several major differences than those based on sinusoids.

1. It is NOT a wave 2. It is NOT continuous 3. It has a range 4. It has asymptotes 4. It does NOT have a period of 2π

However, after considering all of the differences, the same rules apply for determining changes to the period, phase shifts, and vertical shifts.

However, there is NO AMPLITUDE for tangents (but there are vertical stretches).

Tangent vs. Cotangent The graphs of the tangent and cotangent functions are very similar (like the sine and cosine, like the secant and cosecant).

The differences between them are simply a vertical reflection and a horizontal shift.

By remembering a few key aspects of the graphs (their patterns), you can easily and quickly sketch the graphs of tangent and cotangent functions.

Tangent vs. Cotangent – Key Components

At x=0: - The tangent has a “zero.” - The cotangent has an asymptote. - This component repeats after the period. - Half way in between the graph has the “other” characteristic. The tangent goes up (left to right), the cotangent goes down. Half way in between the primary components, the curve has a point the same distance as the amplitude up (or down) from the axis.

Graph:

Graph:

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ASSIGNMENT Unit 4 Assignment 2 – Graphs of Other Trig Functions

- Write out all transformations/characteristics

- Sketch the graph