4.6 Graphs of Other Trigonometric Functions

Graph of Tangent Functions

• Remember tangent is an odd function
• tan(-x) = -tan(x)
• This means that the graph will be symmetric with respect to the origin
• tan x = sin x/cos x
• tan x is undefined when cos x = 0
• cos x = 0 when x=π/2 and x=-π/2
• Thus y=tan x has vertical asymptotes at π/2 and -π/2
• Since we know the period is π there are also asymptotes at x=π/2 + πn where n is an integer
• Period of tangent and cotangent is π/b
• In order to sketch the graph locate key points, identify intercepts and asymptotes, it is usually best to sketch more than one period.

• Period = π
• Domain =  all
• Range = 
• Vertical Asymptotes = 

Graph of Cotangent Functions

• y= cot x = cos x / sin x 
• Cotangent functions are similar to tangent functions however since sin x = 0 makes the cot x undefined the asymptotes are at x = nπ where n is an integer.
• Notice the graphs of tangent are increasing while the graphs of cotangent are decreasing

 

• Period = π
• Domain = all 
• Range =
• Vertical Asymptotes = 

Graphs of Reciprocal Functions

• 1/sin x = csc x
• 1/cos x = sec x 
• These graphs are found by using the reciprocal identities. These graphs also have asymptotes because for sec x for example it is undefined when cos x = 0. 
• Also for given x, the y coordinates of sec x are the reciprocal of the y coordinates of cos x.
• In order to graph the reciprocals csc x and sec x you must first graph the reciprocal functions which are sin x and cos x receptively, keep in mind the asymptotes.
• You should get a u-shaped result, however it is not a parabola.

Cosecant Graph

• Period = 2π
• Domain = all
• Range = 
• Vertical Asymptotes =  
• Symmetry= y-axis 

Secant Graph

 

• Period = 2π
• Domain = all 
• Range = 
• Vertical Asymptotes = 
• Symmetry = origin

If you compare sine and cosine functions to their respective reciprocals you can notice in the graphs that the hills and valleys are interchangeable. For example the hill of the max point on a sine curve is the valley, or low point on a cosecant curve and vice versa.

Shifting Graphs

• To shift the graphs of the parent functions
• Use the same formula
• y = a*tan(b(x-c))+d
• Keep in mind the period for tangent and cotangent is π/b 
• Also amplitude for tangent and cotangent is not necessarily important if the y axis isn't labeled
• Also c is a phase shift and thus also effects asymptotes.
• See section 4.5 for shifts of sin and cosine and definitions of each of the letters

All the graphs in one 

4.7 Inverse Trigonmetric Functions

Section 4.7 explains inverse trigonometric functions as well as compositions of functions.  It is convenient for trigonometric functions to have an inverse because it makes solving for angles of a right triangle much easier, but trig functions are not one-to-one.  By restricting the domain of the trig functions, each function has an inverse.

Inverse Sine

For the y = sin x graph above, the shaded area represents a section of the graph where sin x is one-to-one and the entire range is represented.  The domain [ -π/2, π/2] is where sin x has an inverse function y = arcsin x, or y = sin^-1 x.

The above graph shows y = arcsin x.  Note that the domain and range have switched from the sin x to the arcsin x graphs.  Like all inverse functions, the input- x, and the output- y, switch.  

Inverse Cosine and Inverse Tangent

Inverse Cosine

Similar to y = sin x, y = cos x and y = tan x are not one-to-one unless the domain of these functions are restricted.

For the y = cos x graph the restricted domain is [ 0, π ].  When the x and y values are switched the y =  arccos x graph looks like:

Inverse Tangent

The domain of the y = tan x graph is (-π/2, π/2) to be one-to-one.  Note that unlike the restricted domain of y = sin x with brackets, the restricted domain of y = tan x has parenthesis because y = -π/2 and y = π/2 are vertical asymptotes.

Graph of y = tan x

Graph of y = arctan x

Compositions of Functions

When dealing with an inverse trig function composed of a trig function, or a trig function composed of an inverse trig function, the most important thing to remember is that an inverse trigonometric function equals an angle measurement.  

For example, to solve-

The first step is to recognize that

 

Next, cos x = adjacent/hypotenuse, so 3 is the measure of the adjacent side and 5 is the hypotenuse.  Using the Pythagorean Theorem or special right triangles, the opposite side is found to be 4.  Finally, sin x = opposite/hypotenuse, so 4/5 is the answer.

To sum it up,

Extra

For help with the range of inverse trigonometric functions (the restricted domain of trigonometric functions), here's a table-

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4.4 Trigonometric Functions of Any Angle

Section 4 of chapter 4 involves the trigonometric functions associated with any angle, not just those that are acute.

There are six trigonometric functions: sin (sine), cos (cosine), tan (tangent), csc (cosecant), sec (secant), and cot (cotangent).

These functions represent the relationship between the initial side of the angle and the terminal side.

In standard position, the initial side lies along the positive x-axis and the terminal side lies anywhere and is what defines the angle.

Let θ be an angle in standard position formed by the initial side (x) and the terminal side formed by the point (x,y) when r = √(x²+y²) ≠ 0

sin θ = y/r

cos θ = x/r

tan θ = y/x, x≠0

csc θ = r/y, y≠0

sec θ = r/x, x≠0

cot θ = x/y, y≠0 

As you can see from these relationships, each of the trigonometric functions has their respective reciprocals. The reciprocal pairings are sin θ and csc θ, cos θ and sec θ, and tan θ and cot θ.

e.g 1/sin θ = csc θ and 1/csc θ = sin θ

For each terminal side of an angle, there is a corresponding angle that is called the reference angle. The reference angle is the angle formed by the terminal side of the angle and the x-axis.

(http://calculator.tutorvista.com/reference-angle-calculator.html)

*For angles in the 1st quadrant, the angle is its own reference angle. 

The reason that reference angles are used in trigonometry is that an angle and it's reference angle have equal trigonometric function values, although the sign value of the value can vary. This is useful when evaluating trigonometric functions because you can simplify your value and use the quadrant location to determine the corresponding functions.

EXAMPLE:

The cosine of 7π/6 is ⁻√3/2 and it's reference angle, π/6 has a cosine of √3/2. Since, 7π/6 lies in the 3rd quadrant, we can determine that it's x value (cosine value) is negative.