Cool interactive activities on the web!
Chapter 6. Trigonometry
Do these activities for Trigonometry section 6.2,
and then again for 7.2.
Suggested time committments are in red.
It takes some time
for the java applets to download. Please wait for them.
Section 6.2.
The
sine and cosine activities are highly recommended!
For sine: Play with animated
illustrations
of
1) the
unit circle definition and the graph of sine, (an animated version
of Figure 15 in section 6.2).
Look at how sine isolates
the vertical component of circular motion. (1
minute)
2) the
values of sine
Note: Start with some acute angles (between 0 and 90 degress). But be sure to try other angles, too. You may put in angles larger than 100 degrees (with three or more digits) even though the initial display on some brousers shows only two digits. Just type the angle in and hit start. Be sure to try some large angles and some negative angles. (a few minutes. Play until you understand it.)3) solving the equation "sin x = a". (a few minutes. Play till you get some right.) [If this does not work, the computer probably needs a newer version of java. You can get one at
For cosine: Play with animated
illustrations
of
4) the
unit circle definition and the graph of cosine,
(Notice how cosine isolates the horizontal component of
circular
motion) (1 minute)
5) the
values of cosine
Note: Start with some acute angles (between 0 and 90 degress). But be sure to try other angles, too. You may put in angles larger than 100 degrees (with three or more digits) even though the initial display on some brousers shows only two digits. Just type the angle in and hit "start". Be sure to try some large angles and some negative angles. (a few minutes. Play till you understand it.)6) solving the equation "cos x = a" (a few minutes. Play till you get some right) [This one won't work if the sine one (3) doesn't -- for the same reason. The computer probably needs a newer version of java. You can get one at
For tangent: Play with the
animated
illustration of
7) the
unit circle and the graph of tangent (an animated version of
section 6.2, Figures 35 and 26) (a minute)
(hit the "init" = initialize) button and then "t+' to develop it yourself. Note the scale on the graph is in radians with grid lines at 1, 2, 3 , etc. where it should preferably have grid lines at pi/2, pi, etc.)8) the values of tangent
Note: Start with some acute angles (between 0 and 90 degress). But be sure to try other angles, too. You may put in angles larger than 100 degrees (with three or more digits) even though the initial display on some brousers shows only two digits. Just type the angle in and hit start. Be sure to try some large angles and some negative angles. (a few minutes. Play until you understand it.)
Section 6.3:
For the Law of Sines: A very neat
proof that you create by actively moving triangles!
1) Illustrate
the law of sines with moving triangles!
(Inspect "menu 1". Drag
the red points to make a single triangle. See how a sin B = b sin
A.
"a sin B = b sin A"
yields the Law of Sines immediately by division. Learn how.
Then inspect "menu 2". Drag
its red point.
(several
minutes. Play until you think you could repreduce the argument
for
the Law of Sines, 6.3.3.)
Omit "menu 3.")
Chapter 7. Trigonometry for Calculus.
Section 7.2. Identities
For 7.2 on trig identities, first be sure you have done the sine and
cosine activites from 6.2.
For the Sum-of-Angles formulas: (7.2.11)
1) An interactive version of the figure for
the sine of a sum of angles (an interactive version of 7.2 Figure
9)
Note that sine of the sum is not sum of the sines. (one
minute)
For secant, cosecant, and cotangent:
(7.2.20)
2) An
interactive unit-circle illustration of cotangent, secant, and cosecant
(as well as sine, cosine, and tangent)
(It is best for tangent and cotangent.)
Select tangent and cotangent.
Drag the red point.
Note how, because tangent
is the slope and the y-value where the ray crosses the line x = 1,
then cot x = 1/(tan x) is
the x-value where the ray crosses the line y = 1.
Proof: The ray (line)
at angle t has equation
y = mx = (tan t)x, so
x = y/(tan t), which, when
y = 1, is
x = 1/(tan t) when y = 1.
So,
x = 1/(tan t) = cot t, when
y = 1. (a few minutes)
The idea for secant and
cosecant is not so clear in this java applet. The image in the picture
is giving the reciprocal of the given value (sin t = 1/2
makes
cosecant t = 1/(1/2) = 2), but reciprocals are hard to envision, even
if
they are accurately drawn as in the applet.
Waves. Section 7.4.
The
graph of y = a sin b(x-c). Play with changing the parameters
on the slider bar. Note what increasing b does to the graph. Note how
changing
c shifts the graph left or right, as discussed in section 2.2 on
compostion
of functions. (a few minutes)
Interference
of waves by adding them. Very easy to do.
********
A "short
course" on trig on the web.
The main site on java for mathematics:
http://www.ies.co.jp/math/java/