Graphing the Other Four Trigonometric Functions

Just like the graphs of the sin x and cos x, the graphs of the other four functions have very distinct patterns they can be recognized by. In this blog post we'll be covering the graphs of tan x, cot x, csc, and sec x

These functions also come in the formula        f(x) = a trig[b(x-c)] + d  

The tan x Graph:

To graph the function of tan x, we look back at graphing rational functions to make it easier on us. 

Since tan x equals sin x divided by cos x, we can use that instead. 

By taking each value for the angles in the unit circle and plugging them into the new function, we can find plot-able values for tan x.

When we plot those points, it creates a graph that looks like this. Some important things to remember about the parent tan x graph are:

1. x-intercept at (0,0)

2. Values that make cos x, or the denominator zero, are vertical asymptotes of the graph

3. Values that make sin x, or the numerator zero, are horizontal asymptotes of the graph.

4. The period of tan x is pi

Like sin x and cos x, the value of a vertically stretches and compresses the graph but because the graph approaches infinity when it approaches the asymptotes, you can only notice the difference a makes when compared to another tan x graph. If it is negative however, it will be reflected across the x-axis. 

The value of b affects the period of the graph. If it is negative, it will reflect the graph across the y-axis. This looks exactly like reflecting it across the y-axis however. Unlike sin x and cos x, the formula for finding 

the period or b is: 

 Finally, the values of c and d just shift the graph either to the left or right, or up and down, respectively. Remember that the x within the parentheses needs to have a coefficient of 1. Also within the parentheses, positive means left, negative means right. 

The cot x Graph:

The cot x graph works just like the tan x graph.

So by looking at the table towards the top, we just take the values for cos x and divide them by the values for sin x. This gets us the values for cot x.

Plotting the points gets us the graph:

Important things about the parent cot x graph:

1. x = 0 is a vertical asmyptote 

2. Values that make sin x, or the denominator zero, are vertical asymptotes of the graph

3. Values that make cos x, or the numerator zero, are horizontal asymptotes of the graph.

4. The period of cot x is pi

The values of a, b, c, and d all work the same way they do with the tan x graph. It also uses the formula  

The csc x Graph:

We also make csc x look more like a rational function by doing:

Looking at the table at the top of the graph, we just divide 1 by the values we see for sin x, then we plot. This gets us a graph that looks like this: 

The graphs of csc x and sec x look a little bit different than x and cot x. These graphs make parabola looking things that are boxed in by vertical asymptotes. 

Important things to note about the parent csc x graph:

1. x = 0 is a vertical asymptote 

2. The values of that make sin x equal zero are vertical asmptotes

3. The period is 2 pi

4. The average distance between the minimums and maximums is y = 0

Changing the value of a will change how far away the minimums and maximums are from the "middle line" between. To find the "middle line" of the graph, just add the y values of a minimum and maximum together and divide by 2. If it is negative, the graph reflects across the x-axis. 

Changing the value of b affects the period of the graph. Like the sin x graph, the period for the graph of csc is 2 pi. To find the period, use the formula:

Changing the the value of c shifts the graph to the left or right, but pay attention to the sign of your a and b values so you catch which way the graph is actually moves. 

Changing the value of d shifts the graph up or down. In the case of csc x, it is going to shift the "middle line" up or down. 

The sec x Graph:

Same process, turn sec x into a rational function. So, 

Looking at the table above, we find all the values for sec x and then we plot.

That gets us this graph: 

Important things to note about the parent graph of sec x:

1. x = 0 goes right down the center of the the parabola-like shapes 

2. The values of that make cos x equal zero are vertical asmptotes

3. The period is 2 pi

4. The average distance between the minimums and maximums is y = 0

And finally, the values of a, b, c, and d affect the graph in the same way as they do in the graph of csc x.

So that's nearly everything you need to know about graphing the other four trigonometric functions. Hope this helps! 

4.4 Reference Angles

What is a Reference Angle? 

 - If an angle is in standard position, the reference angle is the acute angle formed by the terminal side of the original angle and the horizontal axis.
- In other words, the reference angle is the other half of the whole or the quadrant.
Examples:

How do you find a Reference Angle?

- You can find the value of an angle from the reference angle, and vice-versa.
- Depending on if the angle is in radians or degrees, you can find the reference angle by subtracting the horizontal axis value (in degrees or radians) from or vice-versa.
-In Quadrant I, the angle is it's own reference angle.
Examples:

  1. = 300° 

Reference Angle= 360° - 300° = 60°

You subtract  from 360° because 360° is the closest x-axis to .

 2. =


The closest x-axis to is the radian value of . Therefore, you have to subtract from because it is in Quadrant III. 

 3.
First, you have to find a co-terminal angle so it is easier to find the reference angle.



Now that you concluded is co-terminal to , you can subtract 150 from 180.


You subtract from 180 degrees because it is the closest x-axis to your co-terminal angle. 


Sources/For More Help:
-Our textbook
-http://www.mathwarehouse.com/trigonometry/reference-angle/finding-reference-angle.php
-http://www.regentsprep.org/Regents/math/algtrig/ATT3/referenceAngles.htm
-http://www.mathopenref.com/reference-angle.html
-http://www.youtube.com/watch?v=3RD-zJUj5Bo&noredirect=1

4.1 Trigonometry: Radian and Degree Measure

What is a radian?
The amount of rotation required such that the length of the intercepted arc is equal to the radius.

Angles 

• Angle-  Two rays with a common endpoint.  Angles are determined by rotating a ray about it's endpoint. 
• Initial Side- The starting position of the ray when measuring an angle
• Terminal Side- The position after the roation of a ray
• An angle can be positive or negative 
• How do we tell if it's positive or negative?  Angles are in standard position, where the vertex is at the origin, and the initial side is at the x-axis

              - Positive angles are generated by counterclockwise rotation
              - Negative angles by clockwise rotation.

• Can be measured in degrees or radians 

Angle Relationships

• Congruent- Angles with equal measures
• Complementary- Angles with measures that add up to 90

  °

   (or π/2 radians)
• Supplementary- Angles with measures that add up to 180

  °

   (or π radians)
• Coterminal- Two angles that when put in stardard position have terminal sides at the same spot

Coterminal angles 

Arc Length 

A circle has a radius of 4 inches.  Find the length of the arc intercepted by a central angle of 240°.

First, convert 240° to radians

240° = 240° ( π radians/180°)

4 π/ 3 radians

s=rθ

= 4(4π/3)

=16π/3 

= about 16.76 inches

Converting degrees to radians 

to convert degrees to radians, multiply degrees by π radians/180°( to cancel out the degrees)

135° =135 * (π radians/180°)= 3π/4 radians 

Converting radians to degrees

to convert radians to degrees, multiply degrees by 180°/π radians (to cancal out the radians) 

2 radians = 2 radians * (180°/ π radians ) = 114.59°