a. Graph y-tanx and y-cotx on the same axes. b. State the domain, the range, and the asymptotes

A. Graph y-tanx and y-cotx on the same axes. b. State the...

Key Concepts

Domain, Range, and Asymptotes

The domain of a trigonometric function consists of all x-values for which the function is defined, which is often restricted due to points of discontinuity caused by division by zero. The range is the collection of y-values that the function can produce, and vertical asymptotes are lines toward which the function tends as it approaches points where it is undefined. These concepts are essential for understanding the complete behavior of functions like tangent and cotangent.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their periodic behavior, identifying points of discontinuity (vertical asymptotes), and noting the cycles of repetition. Key elements include determining the period, amplitude (if applicable), phase shifts, and asymptotic behavior. This process helps in visualizing their behavior over the domain and comparing functions such as tangent and cotangent.

Symmetry and Reflection in Trigonometric Graphs

Symmetry and reflection play a significant role in relating different trigonometric functions. For example, reflecting the graph of the cotangent function about specific lines, such as y = x or y = -x, can yield the graph of the tangent function. These geometric transformations not only demonstrate the inherent relationships between these functions but also provide powerful tools for graph analysis and comparison.

Tangent Function

The tangent function, defined as sin(x)/cos(x), is a periodic trigonometric function with a period of ?. Its graph exhibits vertical asymptotes where cos(x) equals zero, specifically at odd multiples of ?/2, and its range spans all real numbers. The function's behavior is characterized by its increasing nature within each period and its symmetry properties, making its graphical representation distinct among trigonometric functions.

Cotangent Function

The cotangent function, defined as cos(x)/sin(x), is the reciprocal of the tangent function and similarly has a period of ?. Its graph features vertical asymptotes where sin(x) equals zero, which occur at integer multiples of ?, while its range is also all real numbers. The cotangent function shares many characteristics with the tangent function, including periodicity and unbounded behavior, but its asymptote locations and overall orientation differ.

Transcript

00:01 In this question we have to plot y is equal to tan x and y is equal to cot x on the same axis.

00:09 So tan x is equal to cot x implies that x is equal to pi by 4 or 3 pi by 4 or 5 pi by 4 and 7 pi by 4.

00:24 So therefore the value of tan x and cortex when they're equal is equal to 1.

00:31 So when we make the table here we will take x 0 pi by 4, pi by 2, 3 pi by 4, 5 pi 5 5 by 4, 5 5 5 by 2, 7 pi and 2 pi.

00:48 So the value of tan x is equal to 0 1 infinity minus 1 0.

01:00 This is 1 infinity minus 1 0 and the value of cortx which is 1 upon tan x is infinity 1 in 1 0 minus 1 and infinity so let us try to plot this graph so let us assume that this is the line of plus 1 this is the line of minus 1 here tan x is equal to got x values of x are pi by 4 pi by 2 3 pi by 4 5 pi 5 pi 5 5 5 by 4 3 pi 2 7 pi 5 by 2 and 2 pi so for tan x the asymptotes are at pi by 2 at 3 pi by 2 so when we plot tan x we will get tan 0 is equal to 0 10 pi by 4 is 1 it is infinity at pi by 2 tan 3 pi by 4 is minus 1 10 pi is 0 tan 5 5 by 4 is 1 again infinity 7 5 5 by 4 is minus 1 and 2 pi it is 0 so when we join these curves we will get a curve like this now similarly cortx it is infinity at 0 so it has as simple dot at x is equal to 0 at x is equal to pi and at x is equal to 2 pi so at x is equal to 0 it is infinity at x is equal to pi by 4 its value again is 1 at x is equal x is equal to 0 at x is equal to pi by 2 its value is 0 3 pi by 4 it is minus 1 at pi it is infinity at 5 pi by 4 its value is 1.

04:15 At 3 pi by 2 its value is 0.

04:18 At 7 pi by 4 its value is minus 1 and at 2 pi it is infinity.

04:24 So when we join this we will get this curve.

04:33 So the blue one is the cortex curve.

04:36 Y is equal to cortex and the red is y is equal to tan x.

04:41 So therefore in question number d we are being asked that we have to name at least two lines that have the property such that when the graph of cortex is reflected using that line as a reference will result as a graph of tan x.

05:01 So here any line passing through the point where tan x is equal to cortex will result in the reflection of one graph to the other graph.

05:12 So at least two lines are x is equal to 3 pi by 4...

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