WS #2 Introduction to Sine and Cosine
Draw a quick sketch of any graph you are asked to make. If asked to graph something with your
graphing calculator, you can also use Desmos but you need to change things into degrees and change
your window.
We have seen a number of different functions this year (linear, quadratic, cubic, exponential,
logarithmic,…). Each function type has unique characteristics and best fits different situations. We are
now going to look at a new type of functions called trigonometric (trig) functions. There are 6 trig
functions. We are only going to study two of them, sine (sin) and cosine (cos). The other 4 functions
are made up of sine and cosine so they will not be too difficult when you see them in the future years.
The dominant characteristic of trig functions is that they are periodic which means they repeat
themselves. (You can also think of them as waves as you will see when we start to graph them) We
will define sine and cosine using an unit circle. This is a circle of radius 1 centered at (0,0). Since once
we get all the way around a circle things start over, defining sine and cosine with a circle will ensure
that our functions repeat themselves as well.
Cosine is the function that matches an angle on the unit circle with the x coordinate of that location.
So the domain of cosine is an angle (in degrees or radians) and the range is all possible x coordinates
on the unit circle. Sine is the function that matches an angle on the unit circle with the y coordinate of
that location. So the domain of sine is an angle (in degrees or radians) and the range is all possible y
coordinates on the unit circle.
Here are two pictures of what things can look like.  (theta) is the angle and can end up in any
quadrant . (It is a letter of the greek alphabet. It is used as a variable to represent an angle).  can be
in degrees or radians. We will start in degrees first as that is the more familiar way to measure angles.
There is naturally a right triangle always involved in the situation.
3. a) Explain why the domain of the cosine function will be ( , )
b) Explain why the range of cosine will be [-1, 1]
c) Explain why the period of cosine (the length it takes to repeat itself) is 360°
We are now going to graph f(  ) = cos(  ) by hand. The  in the formula will represent the angle we
are at and the f will be the x coordinate at that location.
4. Graph f(  ) = cos(  ) by putting  on the “x” axis and f on the y. Do this for angles between 0° and
360°. This will be one complete loop around the unit circle moving counterclockwise. You can find the
x coordinate exactly at the places where the unit circle hits one of the axis. You can estimate what the
x value is doing for other angles by thinking about how the unit circle looks.
5. Graph y = cosx on your graphing calculator. Put your calculator in degree mode (mode (top left)
and then choose between radians and degrees) and set your x window from 0 to 360 and the y
window from -1 to 1. Cosine is “cos”, two buttons above the “8”. Compare your graphs.
6. Think about the unit circle and its x coordinate at different locations to do the following.
Remember cosine of an angle is the x coordinate at that spot. Solve these without your calculator,
you can use it to check your answers afterwards. Think about angles from 0° to 360°.
a) At what angles is cosine = 0? Explain.
b) For what angles is cosine > 0? Explain.
c) Which is bigger, cos (20°) or cos(30°)? Explain why. Check on your calculator in the end.
d) Which will be negative, cos(100°) or cos(300°)? Explain why. Check on your calculator in the end.
7. Using your calculator, graph y = cosx in a x window of -360 to 360. What is different about this
graph?
8. Using your calculator, graph y = cos(2x) in a x window of 0 to 360. Put this new function in as Y2 so
you can see the “normal” cosine graph at the same time. What is different about this new graph?
9. Using your calculator, graph y = 2cosx in a x window of 0 to 360 and a y window of -2 to 2. Put this
new function in as Y2 so you can see the “normal” cosine graph at the same time. What is different
about this graph?
10. By hand, graph f(  ) = sin(  ) by putting  on the “x” axis and f on the y. Do this for angles
between 0° and 360°. This will be one complete loop around the unit circle moving counterclockwise.
You can find the y coordinate exactly at the places where the unit circle hits one of the axis. You can
estimate what the y value is doing for other angles by thinking about how the unit circle looks.
11. Graph y = sinx on your graphing calculator. Put your calculator in degree mode (mode (top left)
and then degree) and set your x window from 0 to 360 and the y window from -1 to 1. Sine is “sin”
two buttons above the “7”.
12. What is different about the graph of cosine and the graph of sine? What is the same about them?
13. Solve these without your calculator, you can use it to check your answers afterwards.
a) At what angles is sine = 0?
b) For what angles is sine < 0?
c) Which is bigger, sin (20°) or sin(30°)? Explain.
14. a) Find cos (160°) and cos (140°) with your calculator. Use the unit circle to explain why both are
negative. Use the unit circle to explain why 160° gives a bigger negative.
b) Find sin (60°) and sin (120°) with your calculator. Use the unit circle to explain why they both
are positive. Use the unit circle to explain why they are equal.