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! 

Graphing Trigonometric Functions

A periodic function repeats a pattern of y-values (outputs) at regular intervals.

One complete pattern is called a cycle.

The amplitude of y = a sin x and y = a cos x represents half the diasance between the maximum and minimum values of the function and is given by...     Amplitude =  | a |

The sin wave:

The zeros of y = sin x are at the multiples of π.  And it is there that the graph crosses the x-axis, because there sin x = 0.  But what is the maximum value of the graph, and what is its minimum value?

 

Sin x has a maximum value of 1 at π/2, and a minimum value of – 1 at 3 π/2 – and all angels coterminal with them.

Heres the graph of a y= sin(x) 

Properties of the sin function:

Suppose:  y = a sin b θ,   with a ≠0,    b>0,    θ in radians

-          | a | is the amplitude of the function (max/min)

-          b is the number of cycles in the interval from 0 to 2 π

-          2 π/b is the period of the function

-          Graphing a sin function à cut period in half twice

y = sin(x)  (odd function)

-           

- Period = 2 π/b (maxà0minà0àmax) 

-     1 cycle (one entire curve) in 2 π



The cosine wave:

Properties of the cosine function:

Suppose:  y = a cos b θ, with a≠

0, b>0, θ in radians

-        | a | is the amplitude of the function (max/min)

-          b is the number of cycles in the interval from 0 to 2 π

-          2 π/b is the period of the function

y = cos(x)





The graphs of trigonometric functions can be transformed the same way as any other graph...

Consider the function y = a sin [b(x –c)] + d, where a, b, c, and d are constants.  Explain how the value of each of these constants affects the graph of the parent function y = sin x.

 a) Vertically stretches of compresses (can also reflect the x axis if negative)

b) Horizontally stretch of compress

c) moves left or right (phase shift)

d) Shifts up or down (mid-line shift)



Right Triangle Trigonometry

Consider a right triangle, with one of the acute angles labeled as θ, as shown below. Relative to θ, the 3 sides of the triangle are the hypotenuse, the opposite side, and the adjacent side.

  

6 ratios can be formed from this triangle that define the 6 trigonometric functions of the angle θ.

Sine            Cosine            Tangent            Cosecant            Secant            Cotangent

 Note: 0º < θ < 90º and the value of each trigonometric function is positive

         

         

Look! The functions in the second row, are the reciprocals of the functions above them! 

Hint!     SOH            CAH            TOA             CHO            SHA            CAO

Evaluating Trigonometric Functions

Find the 6 trigonometric functions of θ.

 

Find the value of a.

Sketch a right triangle corresponding to the trigonometric function of θ. Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of θ.

                                       

                                       

Other math help sources:

-http://www.regentsprep.org/Regents/math/ALGEBRA/AT2/Ltrig.htm
-http://www.themathpage.com/atrig/solve-right-triangles.htm
-http://www.uiowa.edu/~examserv/mathmatters/tutorial_quiz/trigonometry/righttriangletrig.html
-The textbook
-Mr. Wilhelm!

The Unit Circle

The unit circle is simply a circle that is centered at the origin of a standard
coordinate plane, and has a radius of one

For the unit circle, we also know that half of a rotation is π , and that a full rotation is 2π 

And if we fill in the rest of the unit circle, it looks like this: 

But that's a lot to take in, so let's just focus on the first quadrant.
  

Okay, so where did we get all those points from? Let's think back to something we learned a few years ago. 

At some point in geometry, we learned about these things called.....

SPECIAL TRIANGLES! WOOOOOHOOOOOO

 

But, what does that have to do with the unit circle? Well, if we look back at Quadrant I, the different 
points are at 30°, 45°, and 60°. Coincidence? Probably not.




So, let's take a look at the 30/60/90 triangle first. 

In geometry we learned that we could put in the variable 2s for the hypotenuse, s for the side opposite of 30°, and s√3 for the side opposite of 60°.  

Now let's pretend that this triangle is in the unit circle. That means that the hypotenuse would have to be 1, right? So we set 2s equal to 1 and solve for s. 

 

    2s = 1

__________________

2s/2 = 1/2

__________________

s = 1/2

 

Once we get s, we can plug that in and solve for the other sides, so we end up with:

And if we put this into the unit circle, we would see this: 

The sides of the triangle are the x and y coordinates on the circle at 60°.

The unit circle at 30° makes the same triangle as the one above, it's just flipped and rotated. 

Alright, now the second special triangle is the 45/45 triangle. Geometry taught us that we could put in the variable s√2 for the hypotenuse. And because the 45/45 is an isosceles triangle, we put in s for both sides. 

Again, we want the hypotenuse to equal 1, so we make s√2 = 1 and solve. 

s√2 = 1

__________________

s√2/√2 = 1/√2

__________________

s = (1/√2)  * (√2/√2)

__________________

s = √2/√4 

__________________

s = √2/2

Again, we plug in √2/2 for s, and we get this:

And putting this on the unit circle, we see that the same goes for the 45/45 triangle.

So now we understand the math behind the unit circle YAYYYYYY!