A First Trigonometry Lesson
©2019 Art Middlekauff
For personal home schoolroom use only. Permission must be granted by the author for any other usage. Inspired by Paul A. Foerster’s Algebra and Trigonometry, chapter 13, pp. 711718 (Pearson Prentice Hall, 2006).
Draw this figure on the white board:
Suppose a mouse walked from the center to the edge of the circle (the shaded line). How far will he have walked?
Encourage discussion here. The goal is for the student to say “1 radius.” If the student proposes a normal unit of measure, you can say something like this:
This mouse doesn’t really understand inches, feet, or meters. So let’s just say the mouse remembers the distance he walked and calls it “1 radius.”
So our measuring stick is “1 radius.” If our mouse walked the whole diameter of the circle, how far will he have walked?
Answer: 2 radiuses 

Suppose the mouse walked all the way around the circle. How many radiuses will he travel?
Answer: \(2\pi\) radiuses You may need to review the formula for the circumference of a circle: \(2\pi r\). In this case r is 1 radius, so the circumference is \(2\pi\) radiuses. 

Suppose he only went halfway around the circle. How far will he have gone?
Answer: \(\pi\) radiuses 

How about if he went only a quarter of the way around the circle. How far will he have gone?
Answer: \(\frac{\pi}{2}\) radiuses 

How about \(\frac{1}{8}\) of the way?
Answer: \(\frac{\pi}{4}\) radiuses 

Is there an easier way to express the idea of “walking \(\frac{1}{8}\) of the way along the circle”?
Encourage discussion here. The student may suggest an angle measure. Draw the angle and show it is 45°. 
Let’s make a table:
Portion of circle  degrees  radiuses 
All the way  
Halfway  
\(\frac{1}{4}\) of the way  
\(\frac{1}{8}\) of the way 
Have the student reason and fill in the blank cells as follows:
Portion of circle  degrees  radiuses 
All the way  360°  \(2\pi\) 
Halfway  180°  \(\pi\) 
\(\frac{1}{4}\) of the way  90°  \(\frac{\pi}{2}\) 
\(\frac{1}{8}\) of the way  45°  \(\frac{\pi}{4}\) 
Let’s do some mental math. Don’t look at the table.
How many radiuses is 360°?
How many radiuses is 45°?
How many radiuses is 90°?
How many radiuses is 180°?
Do you like the word “radiuses”? Can you think of a better word?
In 1873, Lord Kelvin’s little brother was writing exam questions. He didn’t really like the word “radiuses” either, so he coined a new word: a radian.
Update the table and change the “radiuses” column label to “radians.”
So how many radians is it for our mouse to walk around the entire circle?
Answer: \(2\pi\) radians
How many radians are equal to 180°?
Answer: \(\pi\) radians
How many radians are equal to 90°?
Answer: \(\frac{\pi}{2}\) radians
Now suppose we wanted to track the position of the mouse as he travels around the circle. Let’s say that the circle has a radius of 1 unit. We call this the “unit circle.”
Add tick marks for x and y at 1 and 1:
Let’s track his position using x and y coordinates:
Portion of circle  degrees  radians  x  y 
All the way  360°  \(2\pi\)  
Halfway  180°  \(\pi\)  
\(\frac{1}{4}\) of the way  90°  \(\frac{\pi}{2}\) 
Have the student reason and fill in the first three rows of blank cells as follows:
Portion of circle  degrees  radians  x  y 
All the way  360°  \(2\pi\)  1  0 
Halfway  180°  \(\pi\)  1  0 
\(\frac{1}{4}\) of the way  90°  \(\frac{\pi}{2}\)  0  1 
We call the xcoordinate on the unit circle “cosine” and the ycoordinate “sine.” The name sine comes from the Arabic word for bowstring. The yvalue of the position of the mouse resembles half of a bowstring:
The prefix “co” in cosine means “complement.”
Update the table and change the “x” column label to “cosine” and the “y” column label to “sine”:
Portion of circle  degrees  radians  cosine  sine 
All the way  360°  \(2\pi\)  1  0 
Halfway  180°  \(\pi\)  1  0 
\(\frac{1}{4}\) of the way  90°  \(\frac{\pi}{2}\)  0  1 
What is the sine of \(\frac{\pi}{2}\) radians? In other words, what is the ycoordinate of the mouse when he has gone \(\frac{\pi}{2}\) radiuses around the circle?
Answer: 1
Mental math:
What is the cosine of \(\pi\)?
Answer: 1
What is the sine of 0?
Answer: 0
What is the sine of \(\frac{3\pi}{2}\)?
Answer: 1
People tend to think of geometry and algebra are two different kinds of math. But when you get to calculus, you will see that they are very closely related. This is especially true when you use radians. There is something special about radians that makes some elements of calculus very simple and elegant. That is why most mathematicians prefer to measure angles in radians instead of degrees.