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Review : Trig Functions

The intent of this section is to remind you of some of the more important (from a Calculus standpoint…) topics from a trig class.  One of the most important (but not the first) of these topics will be how to use the unit circle.  We will actually leave the most important topic to the next section.

 

First let’s start with the six trig functions and how they relate to each other.

 

 

 

 

Recall as well that all the trig functions can be defined in terms of a right triangle.

Right_Triangle

 

From this right triangle we get the following definitions of the six trig functions.

 

 

 

 

 

 

 

 

Remembering both the relationship between all six of the trig functions and their right triangle definitions will be useful in this course on occasion.

 

Next, we need to touch on radians.  In most trig classes instructors tend to concentrate on doing everything in terms of degrees (probably because it’s easier to visualize degrees).  The same is true in many science classes.  However, in a calculus course almost everything is done in radians.  The following table gives some of the basic angles in both degrees and radians.

 

Degree

0

30

45

60

90

180

270

360

Radians

0

 

 

 

 

 

 

 

 

Know this table!  We may not see these specific angles all that much when we get into the Calculus portion of these notes, but knowing these can help us to visualize each angle.  Now, one more time just make sure this is clear.

 

Be forewarned, everything in most calculus classes will be done in radians!

 

Let’s next take a look at one of the most overlooked ideas from a trig class.  The unit circle is one of the more useful tools to come out of a trig class.  Unfortunately, most people don’t learn it as well as they should in their trig class. 

 

Below is the unit circle with just the first quadrant filled in.  The way the unit circle works is to draw a line from the center of the circle outwards corresponding to a given angle.  Then look at the coordinates of the point where the line and the circle intersect.  The first coordinate is the cosine of that angle and the second coordinate is the sine of that angle.  We’ve put some of the basic angles along with the coordinates of their intersections on the unit circle.  So, from the unit circle below we can see that  and

 

Unit_Quarter

Remember how the signs of angles work.  If you rotate in a counter clockwise direction the angle is positive and if you rotate in a clockwise direction the angle is negative.

 

Recall as well that one complete revolution is , so the positive x-axis can correspond to either an angle of 0 or  (or , or , or , or , etc. depending on the direction of rotation).  Likewise, the angle  (to pick an angle completely at random) can also be any of the following angles:

 

 (start at  then rotate once around counter clockwise)

 

 (start at  then rotate around twice counter clockwise)

 

 (start at  then rotate once around clockwise)

 

 (start at  then rotate around twice clockwise)

 

etc.

 

In fact  can be any of the following angles  In this case n is the number of complete revolutions you make around the unit circle starting at .  Positive values of n correspond to counter clockwise rotations and negative values of n correspond to clockwise rotations.

 

So, why did I only put in the first quadrant?  The answer is simple.  If you know the first quadrant then you can get all the other quadrants from the first with a small application of geometry.  You’ll see how this is done in the following set of examples.

 

Example 1  Evaluate each of the following.

(a)  and    [Solution]

(b)  and    [Solution]

(c)  and    [Solution]

(d)    [Solution]

Solution

(a) The first evaluation in this part uses the angle .  That’s not on our unit circle above, however notice that .  So  is found by rotating up  from the negative x-axis.  This means that the line for  will be a mirror image of the line for  only in the second quadrant.  The coordinates for  will be the coordinates for  except the x coordinate will be negative.

 

Likewise for  we can notice that , so this angle can be found by rotating down  from the negative x-axis.  This means that the line for  will be a mirror image of the line for  only in the third quadrant and the coordinates will be the same as the coordinates for  except both will be negative.

 

Both of these angles along with their coordinates are shown on the following unit circle.

Unit_Quarter_With_Angles

From this unit circle we can see that  and .

 

This leads to a nice fact about the sine function.  The sine function is called an odd function and so for ANY angle we have

 

[Return to Problems]

 

(b) For this example notice that  so this means we would rotate down  from the negative x-axis to get to this angle. Also  so this means we would rotate up  from the negative x-axis to get to this angle.  So, as with the last part, both of these angles will be mirror images of  in the third and second quadrants respectively and we can use this to determine the coordinates for both of these new angles.

 

Both of these angles are shown on the following unit circle along with appropriate coordinates for the intersection points.

 

Unit_Quarter_With_Angles

From this unit circle we can see that  and .  In this case the cosine function is called an even function and so for ANY angle we have

.

[Return to Problems]

 

(c) Here we should note that  so  and  are in fact the same angle!  Also note that this angle will be the mirror image of  in the fourth quadrant.  The unit circle for this angle is

 

Unit_Quarter_With_Angles

Now, if we remember that  we can use the unit circle to find the values of the tangent function.  So,

 

.

 

On a side note, notice that  and we can see that the tangent function is also called an odd function and so for ANY angle we will have

.

[Return to Problems]

 

(d) Here we need to notice that .  In other words, we’ve started at  and rotated around twice to end back up at the same point on the unit circle.  This means that

                                            

Now, let’s also not get excited about the secant here.  Just recall that

                                                            

and so all we need to do here is evaluate a cosine!  Therefore,

                                     

[Return to Problems]

 

So, in the last example we saw how the unit circle can be used to determine the value of the trig functions at any of the “common” angles.  It’s important to notice that all of these examples used the fact that if you know the first quadrant of the unit circle and can relate all the other angles to “mirror images” of one of the first quadrant angles you don’t really need to know whole unit circle.  If you’d like to see a complete unit circle I’ve got one on my Trig Cheat Sheet that is available at http://tutorial.math.lamar.edu.

 

Another important idea from the last example is that when it comes to evaluating trig functions all that you really need to know is how to evaluate sine and cosine.  The other four trig functions are defined in terms of these two so if you know how to evaluate sine and cosine you can also evaluate the remaining four trig functions.

 

We’ve not covered many of the topics from a trig class in this section, but we did cover some of the more important ones from a calculus standpoint.  There are many important trig formulas that you will use occasionally in a calculus class.  Most notably are the half-angle and double-angle formulas.  If you need reminded of what these are, you might want to download my Trig Cheat Sheet as most of the important facts and formulas from a trig class are listed there.

 

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