4.3 Graphs of the Trigonometric Functions

 

From the previous section, we can construct a table of values of the trigonometric functions for the multiples of

π / 4, or 45o, between 0 and 2π inclusive.

 

 

Exercise 4.3.1

 

Using , sketch the graph of y = sin x between x = 0 and x = 2 π.

 

Solution

 

Exercise 4.3.2

 

Using , sketch the graph of y = cos x between x = 0 and x = 2 π.

 

Solution

 

Exercise 4.3.3

 

Construct a table of values of sin x between x = 2 π and x = 4 π in increments of π / 4. 

Sketch the graph of y = sin x between x = 0 and x = 4 π.

 

Solution

 

Notice that the sine graph between x = 2 π and x = 4 π repeats the graph between x = 0 and x = 2 π.  The graph of the sine function consists of infinite repetitions of the portion of the graph between x = 0 and x = 2 π.  The sine graph, as well as the other trigonometric functions, satisfy the property that f ( x + 2 π) = f ( x ) for all x in the domain of the function.  This is because the angle x and the angle x + 2 π in standard position have the same terminal side.  Functions with the property that, for some constant d, f ( x + d ) = f ( x ) for all x in the domain of the function are called periodic functions.  For a periodic function f, the smallest value d for which

f ( x + d ) = f ( x ) for all x in the domain of f is called the period of f.  The sine and cosine functions are periodic functions with period 2 π.

 

Notice that the graph of cosine looks like the graph of sine shifted π / 2 units to the left.  In other words,

cos x = sin ( x + π / 2 ).

 

Now we will investigate the graphs of the reciprocal functions of sine and cosine—the graphs of cosecant and secant.

 

csc x = 1 / sin x and sec x = 1 / cos x .

 

Whenever sin x = 1, then csc x = 1.  Whenever cos x = 1, then sec x = 1.

 

Whenever sin x = -1, then csc x = -1.  Whenever cos x = -1, then sec x = -1.

 

Whenever sin x = 0, csc x is undefined, and the graph of cosecant has a vertical asymptote.

 

Whenever cos x = 0, sec x is undefined, and the graph of secant has a vertical asymptote.

 

Exercise 4.3.4

 

Using , sketch the graph of y = csc x between x = 0 and x = 2 π.  Sketch the vertical asymptotes as dotted lines.

 

Solution

 

Exercise 4.3.5

 

Sketch the graph of  y = sec x between x = 0 and x = 2 π.  Sketch the vertical asymptotes as dotted lines.

 

Solution

 

Next, we will consider the graphs of tangent and cotangent.

 

Recall that tan x = ( sin x ) / ( cos x ).  The tangent graph has a vertical asymptote anywhere cosine equals zero.

 

Exercise 4.3.6

 

Sketch the graph of  y = tan x between x = 0 and x = 2 π.  Sketch the vertical asymptotes as dotted lines.

 

Solution

 

Notice that the graph of tangent between x = π and x = 2 π is a repeat of the graph of tangent between x = 0 and x = π.  The period of tangent, unlike sine, cosine, cosecant and secant, is π rather than 2 π.  Thus

 

tan ( x + π ) = tan x for all x in the domain of the tangent function.

 

The tangent function of an angle A equals the slope of the terminal side of A when A is in standard position.  If one adds 180o, or π radians to A, the result will be an angle whose terminal side has the same slope as the terminal side of A.

 

The cotangent is the reciprocal of the tangent function.

 

cot x = 1 / tan x

 

Whenever tan x = 1, cot x = 1.  Whenever tan x = -1, cot x = -1. 

 

Whenever tan x = 0, cotangent is undefined and its graph has a vertical asymptote.

 

Whenever tangent is undefined, cotangent equals zero.

 

Exercise 4.3.7

 

 

Sketch the graph of  y = cot x between x = 0 and x = 2 π.  Sketch the vertical asymptotes as dotted lines.

 

Solution

 

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