If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
Precalculus II
Asst # 7
Self-critique (if necessary):
Self-critique Rating:
**** Query problem 6.6.12
cot(2`theta/3) = -`sqrt(3)
Your solution:
Confidence Assessment:
Given Solution:
The cotangent function takes value sqrt(3) when its argument is 5 pi / 6 or 11 pi / 6, as can easily be seen using a labeled unit circle and the definition of the cotangent.
If the argument of the function is coterminal with 5 pi / 6 or 11 pi / 6 we obtain the same value for the cotangent.
So the possible arguments are 5 pi / 6 + 2 pi n or 11 pi / 6 + 2 pi n, where n can be any integer.
The student solution given below is correct for arguments 5 pi / 6 and 11 pi / 6, but does not address possible coterminal arguments.
2`theta/3 = 5pi/6 or 11pi/6
`theta = 5pi/6 * 3/2 or 11pi/6 * 3/2
`theta = 5pi/4 or 11pi/4
.........................................
23:43:44
......!!!!!!!!...................................
Self-critique (if necessary):
Self-critique Rating:
**** List all the
possible values of 2 `theta / 3 such that the equation is satisfied (the list is
infinite; use the ellipsis ... to indicate the continuation of a pattern)
......!!!!!!!!...................................
Your solution:
Confidence Assessment:
Given Solution:
The cotangent function is periodic with period pi. Thus if any integer multiple of pi is added to 2`theta/3, we still have a solution.
So our solutions have the form 2/3 theta + n * pi = 5 pi / 4, where n can be any integer.
We can subtract the n * pi from both sides to get
2/3 theta = 5 pi / 4 - n * pi.
Noting that n can be a positive or negative integer, we get all the same solutions if we change the sign of n, so that our solutions could be written
2/3 theta = 5 pi / 4 + n * pi.
For example, when n = 2 we would get the equation
2/3 theta = 5 pi / 4 + 2 * pi
Writing the right-hand side with common denominator 4 we have
2/3 theta = 5 pi / 4 + 8 pi / 4 = 13 pi / 4
so that
theta = 3/2 * 13 pi / 4 = 39 pi / 8.
We could individually substitute values 0, 1, 2, 3, ... for n, and would obtain the following solutions:
For n = 0, theta = 15 pi / 8.
For n = 1, theta = 27 pi / 8.
For n = 2, theta = 39 pi / 8.
For n = 3, theta = 51 pi / 8.
For n = 4, theta = 63 pi / 8.
For n = 5, theta = 75 pi / 8.
For n = 6, theta = 87 pi / 8.
Alternatively we could solve the equation
2/3 theta = 5 pi / 4 + n * pi
for theta, obtaining
theta = 3/2 ( 5 pi / 4 + n pi) = 3/2 ( 5 pi / 4 + 4 n pi / 4) = 3/2 (5 + 4 n) * pi / 4 = (15 + 12 n pi) / 8.
and substitute n values 0, 1, 2, 3, ... .
We would obtain
the same values of theta as before.
.........................................
23:57:51
......!!!!!!!!...................................
Self-critique (if necessary):
Self-critique Rating:
**** How many of these
values result in `theta values between 0 and 2 `pi?
......!!!!!!!!...................................
23:59:12
Your solution:
Confidence Assessment:
Given Solution:
From preceding calculations, we see that n = 0 yields theta = 15 pi / 8, which is less than 2 pi. It is also clear from preceding results that no positive value of n gives us a solution between 0 and 2 pi.
We could begin finding solutions for negative values of n, and if so we would quickly see that n = -1 gives us a solution between 0 and 2 pi, but no other negative value of n does so.
A more powerful method, for which we would be grateful if there were a large number of such solutions, is as follows:
We know from before that our solutions are of the form
theta = (15 + 12 n) pi / 8, for integer values of n.
0 <= theta < 2 pi means
0 <= (15 + 12 n ) pi / 8 < 2 pi.
Multiplying all expressions by 8 we get
0 <= (15 + 12 n) pi < 16 pi.
Dividing both sides by pi we have
0 <= 15 + 12 n < 16 so that
-15 <= 12 n < 1 and
-5/4 < n < 1/12.
So n is an integer between -5/4 and 1/12.
The only integers that satisfy this are -1 and 0.
So n = -1 and n = 0 give us values of theta between 0 and 2 pi.
These values are found by substituting 0 and -1 into our solution for theta, obtaining
theta = 15 pi / 8 (for n = 0) and
theta = 3 pi / 8 ( for n = -1).
.........................................
Self-critique (if necessary):
Self-critique Rating:
**** Query problem 6.6.44
solve sin^2(`theta) = 2 cos(`theta) + 2
......!!!!!!!!...................................
14:41:12
Your solution:
Confidence Assessment:
Given Solution:
** Since sin^2(`theta) = 1
- cos^2(`theta) we have
1 - cos^2(`theta) = 2 cos(`theta) + 2. This equation is a quadratic equation in
cos(`theta). To see this rearrange the equation to get
cos^2(`theta) + 2 cos(`theta) + 1 = 0. Now let u = cos(`theta). You get
u^2 + 2 u + 1 = 0. This is a quadratic equation with solution u = -1.
Thus our solution is u = -1, meaning cos(`theta) = -1, so `theta = -pi + 2 `pi
k, for any integer k. **
.........................................
14:43:04
......!!!!!!!!...................................
Self-critique (if necessary):
Self-critique Rating:
**** Query problem
6.6.66 19x + 8 cos(x) = 0
Your solution:
Confidence Assessment:
Given Solution:
You can't get a
closed-form solution, but you can graph y = 8 cos(x) and y = -19 x and use the
graph to estimate the value of x where the two are equal.
You should be able to construct the graphs of the two functions and make a
reasona ble estimate.
If you have constructed these graphs you should be able to understand the
following:
8 cos(x) goes from 0 to 8 as x goes from -pi/2 to 0. -pi/2 is about -1.57.
-19 x = 8 when x = -8/19, or about -.42.
So the graph of -19 x is higher than the highest value of cos(x) before we get
much more than 1/4 of the way from x = 0 to x = -pi/2.
It's easy then to conclude that the graphs cross a little ways before we get to
x = -.42, as we move to the left from the y axis.
Using computer algebra software we can refine the result to any number of
significant figures. To three significant figures the solution is x = -.390
.........................................
Self-critique (if necessary):
Self-critique Rating: