GSP ACTIVITY: THE SINE AND COSINE RATIOS
We’re going to need more accuracy to our ratio calculations today than usual. So, when you open your
new sketch, go to Edit and then Preferences. You can leave the Angle and Distance measurements at
“hundredths”, but change the Other (Slope, Ratio,…) to “ten-thousandths” and then click OK.
Our goal today is to investigate the relationship between an angle in a right triangle and two ratios
associated with that angle.
First, we need to create a right triangle that allows us to “neatly” adjust the angles. Here is how we do
that:
1.
2.
3.
4.
5.
Create a horizontal segment near the bottom of your sketch. If you press the shift key while
you drag to make the segment, GSP will “lock” it into being horizontal. Label the left endpoint
as A and the right endpoint as B.
Highlight the right endpoint and the segment. Then, use the construct menu to make a
perpendicular line.
Plot a point P on the horizontal segment.
Select just point P. From the transform menu, choose Translate and move the point by 10 cm at
a 90 degree angle. This will create a new point P’.
Construct a circle using the point P on the horizontal segment as the center and then locking the
control point of the circle onto the translated point P’. You may need to slide point P over so
that your circle intersects the perpendicular line. Your image should look as follows:
P'
A
P
B
6. Plot a point where the circle intersects the perpendicular line and label this as point K. Then,
draw a segment from the point P to point K. Now your image should look as follows:
P'
K
A
P
B
7. Hide the circle, hide point P’, hide the perpendicular line, hide point A and segment AB. (But, do
not hide point B nor point C nor point K nor segment PK.)
8. Now, draw segments from K to B and from B to P. In the end, you image should look as follows:
K
P
9.
10.
11.
12.
13.
B
Now, drag point P back and forth and you will see that you have a nicely adjustable right
triangle.
Drag P so that angle P is very small. Highlight, in order, points K, P and B and then find the
measure of this angle (from the measure menu). Drag the label over near your angle. If you
want to make the text bigger, highlight just the measurement and then choose Text from the
Display menu and then select Increase Size. Continue until it is as large as you like.
Measure all three sides of your triangle and drag the measurements near to their sides.
Now, use the Calculate option from the Measure menu to evaluate the ratio of BK to PK. (To do
this, after choosing the calculate option, you will see a pop-up calculator. Just click on the BK
measurement in your sketch and then the division symbol, and then the PK measurement in
your sketch, then OK. You will see the ratio and its value appear in your sketch. )
Note, BK is opposite angle P and PK is the hypotenuse. So, let’s create a text box and type in
“Ratio of Opposite to Hypotenuse”. Drag this near to the ratio calculation. And drag the ratio
and the textbox over near your angle. At this point, your sketch should looks as follow (though
your measures will be a bit different:
m PK = 10.00 cm
mKPB = 4.02
P
m PB = 9.98 cm
K
B
m BK = 0.70 cm
Ratio of opposite to
hypotenuse:
m BK
m PK
= 0.0701
14. Now, use the calculate menu to measure the ratio of PB to PK.
15. Note that this is the ratio of the side adjacent to angle P and the hypotenuse. So, create a text
box that says “Ratio of adjacent to hypotenuse:” and drag it near to this new ratio. You may
want to position the new textbox and ratio under the other ratio and measurement as shown:
m PK = 10.00 cm
mKPB = 4.02
P
m PB = 9.98 cm
K
B
m BK = 0.70 cm
Ratio of opposite to
hypotenuse:
m BK
m PK
= 0.0701
Ratio of adjacent to
hypotenuse:
m PB
m PK
= 0.9975
16. Enter the numbers into the first row of your table. (Of course, the ratio of opposite to
hypotenuse has a name. It is called the “sine” of the angle. And, the ratio of adjacent to
hypotenuse is called the “cosine” of the angle.)
17. Now, from document options (under the file menu), choose to add a new page and make it a
duplicate of page 1. You will see a tab appear along the bottom of your sketch. You are now on
page 2 of your sketch (which at the moment is identical to page 1).
18. Drag point P to make an angle approximately 20 – 25 degrees. Reposition your measurements
so that they do not overlap the triangle. Record the three values into the second row of your
table.
19. Now, from document options (under the file menu), choose to add a new page and make it a
duplicate of page 2. You will see a tab appear along the bottom of your sketch. You are now on
page 3 of your sketch (which at the moment is identical to page 2).
20. Drag point P to make an angle approximately 45-50 degrees. Reposition your measurements so
that they do not overlap the triangle. Record the three values into the third row of your table.
21. Now, from document options (under the file menu), choose to add a new page and make it a
duplicate of page 3. You will see a tab appear along the bottom of your sketch. You are now on
page 4 of your sketch (which at the moment is identical to page 3).
22. Drag point P to make an angle approximately 65-70 degrees. Reposition your measurements so
that they do not overlap the triangle. Record the three values into the fourth row of your table.
23. Repeat one more time (adding a duplicate page) and record another row of your table for an
angle near 90 degrees.
You now have a table that serves as a great summary of our findings today! Furthermore, you have the
visuals to go along with it. Given a size of an angle, we should in our mind be able to estimate the sine
and cosine of an angle just by thinking about the ratios!
Finally, there is one more relationship to point out! See the reverse side of your table!
A RELATIONSHIP BETWEEN SINE AND COSINE:
Using the measurements below, find the following values. Show your work and round your answer to
three decimal places.
a)
Sine of 30 degrees:
b) Cosine of 60 degrees:
c) Sine of 60 degrees:
d) Cosine of 60 degrees:
e) Try to draw a verbal conclusion about the relationship between the sine and cosine.
f)
Now, try to use symbols to describe this relationship. (You might want to introduce a variable X
for an angle measure.)
mABC = 60.00
B
m AB = 15.70927 cm
m BC = 7.85464 cm
A
mBAC = 30.00
m AC = 13.60463 cm
C
SINE AND COSINE RECORDING SHEET:
Angle
Picture
Ratio of Opposite
to Hypotenuse
(Sine of the Angle)
Ratio of Adjacent
to Hypotenuse
(Cosine of the
Angle)