#$&*
course EGR 140
Time and Date Stamps (logged): 12:14:20 02-11-2012 °±Ÿ°³Ÿ±¯¯±Ÿ°°Ÿ±¯°±Precalculus II Test 1
________________________________________
Completely document your work and your reasoning.
You will be graded on your documentation, your reasoning, and the correctness of your conclusions.
________________________________________
Test should be printed using Internet Explorer. If printed from different browser check to be sure test items have not been cut off. If items are cut off then print in Landscape Mode (choose File, Print, click on Properties and check the box next to Landscape, etc.).
Signed by Attendant, with Current Date and Time: ______________________
If picture ID has been matched with student and name as given above, Attendant please sign here: _________
Instructions:
• Test is to be taken without reference to text or outside notes.
• Graphing Calculator is allowed, as is blank paper or testing center paper.
• No time limit but test is to be taken in one sitting.
• Please place completed test in Dave Smith's folder, OR mail to Dave Smith, Science and Engineering, Va. Highlands CC, Abingdon, Va., 24212-0828 OR email copy of document to dsmith@vhcc.edu, OR fax to 276-739-2590. Test must be returned by individual or agency supervising test. Test is not to be returned to student after it has been taken. Student may, if proctor deems it feasible, make and retain a copy of the test..
Directions for Student:
• Completely document your work.
• Numerical answers should be correct to 3 significant figures. You may round off given numerical information to a precision consistent with this standard.
• Undocumented and unjustified answers may be counted wrong, and in the case of two-choice or limited-choice answers (e.g., true-false or yes-no) will be counted wrong. Undocumented and unjustified answers, if wrong, never get partial credit. So show your work and explain your reasoning.
• Due to a scanner malfunction and other errors some test items may be hard to read, incomplete or even illegible. If this is judged by the instructor to be the case you will not be penalized for these items, but if you complete them and if they help your grade they will be counted. Therefore it is to your advantage to attempt to complete them, if necessary sensibly filling in any questionable parts.
• Please write on one side of paper only, and staple test pages together.
Test Problems:
. . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
Problem Number 1
Develop a formula for sin( 11 `theta) as a degree- 11 polynomial in cos(`theta).
.
.
.I am not sure how to do this one.
.
.
.
.
.
.
.
Problem Number 2
The average monthly temperatures at a certain location are 31.10182, 35.24695, 44.41197, 56.14355, `y5, 74.89857, 76.9016, 72.77423, 63.62131, 51.89292, 40.72856 and 33.11675 degrees in months 1, 2, 3, …, 12 of the year. Give a model of the form y = A sin(`omega t - `phi) + c for these temperatures.
.
.
.So for this one I would make a table first.
1 31.10182
2 35.24695
3 44.41197
4 56.14355
5 `y5
6 74.89857
7 76.9016
8 72.77423
9 63.62131
10 51.89292
11 40.72856
12 33.11675
.
.And then honestly I don’t know where to go from here. I would find the amplitude right? But the `y5 is throwing me off. I don’t really know where to go from here. Can you give me step by step how to do this?
.
.
.
@&
The `y5 is just a variable that didn't get filled in with a number. Simply ignore that line. It doesn't seriously affect the trend of the data.
What is the greatest y value?
What is the least?
What therefore is the y value in the middle?
How long does it take the data to complete a full cycle?
If y = A sin(omega t - phi) + c hen
What is the greatest y value?
What is the least?
What is the y value in the middle?
How long does a cycle last?
From your answers to all the above questions:
What can you conclude about the values of A and c appropriate to the given data? What can you conclude about omega?
*@
.
.
Problem Number 3
Give the y-intercepts of all six trigonometric functions. In terms of the definition of each function, explain why the y intercept is where it is.
.
.For sin(x) the y intercept is 0. It is an odd function with symmetry along its 0,0 origin.
For cox(x) the y intercept is at 1. It
For tan(x) the y intercept is 0.
For cot(x) there are no y intercepts. It is an exponential graph.
For sec(x) the y intercept is at 1.
For csc(x) there are no y intercepts.
Am I right with all of this? I’m really not sure how to explain why the y intercepts are where they are at.
.
.
.
.
.
.
@&
Your answers are generally good.
None of the graphs are exponential, but the cotangent and cosecant are asymptotic to the y axis and as you say do not have y intercepts.
*@
.
.
Problem Number 4
The tire of an automobile completes 12 revolutions in a second. The diameter of the tire is 20 inches. How fast is the rim of the tire moving?
12 rev/sec=12*2pi10
=62.8 rad/sec.
.
I am not sure if I have done this correctly or not.
.
.
.
@&
The speed of the tire is measured in distance units per second; in this case that could be inches / second.
The radian is a measure of angle, not a measure of distance.
For a circle of radius 10 inches, a radian of angle would correspond to 10 inches of arc along the circumference.
A revolution is 2 pi radians.
You've got the right numbers in your calculation but you need to use the right units (you also appear to have neglected to multiply by 12).
The correct calculation, with units, would be
12 rev / second * 2 pi radians / revolution * 10 inches / radian = 240 pi inches / second, which could be approximated as about 750 inches / second.
In the calculation the revolutions divide out, the radians divide out and you're left with inches in the numerator and seconds in the denominator. So your result is in units of inches / second.
*@
.
.
.
Problem Number 5
Determine the phase shift, amplitude and period of the function y = -2 cos( 4 `pi t + `pi / -4 ).
.The amplitude is -2, the phase shift is (pi/-4)/(4pi). The period would be 2pi/4pi or pi/2pi.
I think that I have this correctly.
.
.
.
.
.
.
@&
Good.
You would also want to simplify your results.
(pi / (-4)) / (4 pi) = -1/16.
2 pi / (4 pi) = 1/2.
*@
.
.
Problem Number 6
If an arc of a circle of radius 10 is subtended by a central angle of 13 radians, then what is the length of the arc?
.
.
.Okay, for this one, I need to use my unit circle correct? I’m just not sure where to start. Is it similar to the automobile one?
.
.
.
.
.
.
@&
Every radian corresponds to an arc distance equal to the radius.
So 13 radius would correspond to an arc distance equal to 13 times the radius, or 13 * 10 = 130.
*@
.
Problem Number 7
Give the period, domain and range of each of the six trigonometric functions.
.
.For sin(x) the period is 2pi, the domain is all real numbers, and the range is (-1,1)
For cos(x) the period is 2pi, the domain is all real numbers and the range is (-1,1)
For tan(x) the period is pi, the domain is all real numbers, and the range is all real numbers
For cot(x) the period is pi, the domain is all real numbers except kpi, the range is all real numbers
For sec(x) the period is 2pi, the domain is all real numbers except pi/2+kpi, the range is -infinity , -1] U [1 , +infinity)
For csc(x) the period is 2pi, the domain is all real numbers except kpi, the range is (-infinity , -1] U [1 , +infinity)
.
.
.
.
.
.
.
.
.
.
.
.
.
Problem A: Explain in detail how we derive the values of the trigonometric functions of 45 degrees using a 45-45 right triangle with a leg of length 1. Explain how we know the lengths of two the sides from basic geometry, and how we then use the Pythagorean Theorem to find the length of the third side. Then explain in terms of the definition of the sine function how we use this triangle to find the exact value of the cotangent of 45 degrees.
.We know that a right triangle has a 90 degree angle and that the two angles are equal so we could say 1^2+1^2=sqrt(2).
I am not sure how you would use the sine function?.
.
.
.
.
.
.
.
.
@&
From your results you can conclude that the point (1, 1) lies on a circle of radius sqrt(2), centered at the origin.
For the point (1, 1) on that unit circle, r = sqrt(2), x = 1 and y = 1.
The circular definition of the trigonometric functions includes sin(theta) = y / r, cos(theta) = x / r, etc..
*@
.
.
.
.
Problem B: Sketch the unit circle, indicate all angles which are multiples of pi/4 and indicate the x and y coordinates of the unit-circle point corresponding to each multiple. Use these values to make a table of sin(theta) vs. theta, and a table of cos(theta) vs. theta. Sketch on a single set of clearly labeled coordinate axes the graph of each function. Be sure the horizontal-axis intercepts of the graph are labeled.
.
. 0, pi/4, pi/2, 3 pi/4, pi, 5 pi/4, 3 pi/2, 7 pi/4, and 2 pi.
This problem is greatly confusing me. To find the angles, I would do what?
Then I need to make a table with what numbers on one side? I would then take those numbers and put them into a formual correct?
.
.
.
.
@&
You've listed 9 angles, which break the interval from 0 to 2 pi into 8 equal intervals.
A circle centered at the origin covers an angle of 2 pi radians.
So your intervals break the circle into 8 equal sectors.
You would sketch the circle, divide it into 8 equal angles, and label the angles 0, pi/4, pi/2, 3 pi/2, etc..
Your table would include these angles as the first column.
Two additional columns would consist of the corresponding values of the sine and cosine.
*@
.
.
.
.
.
.
.
.
.
Problem C: A certain town is 528 miles north and in unspecified distance east or west, and ends up 640 miles from an airport. In what direction, as measured counterclockwise from the easterly direction, did the airplane fly?
.
.
.
.I would use the Pythagorean theorem to find the radius.
So 640^2=528^2+x^2
=361.7
So this is my radius. I would then do 528/361.7=
.
.0.99 and then the inverse of this is going to be what my direction will be correct?
I am not sure about this one at all. IF my number is positive though that will be east and if it is negative it will be west right? Or how does this work? Help please?
.
@&
Good.
You want to do arctan(528 / 361.7)
That will come out close to 1 radian, so your .99 probably indicates .99 radian. This would be around 59 degrees, indicating 59 degrees north of east.
Your equation 528^2 + x^2 = 640^2 also has solution x = -361.7, so another possible angle is arcTan(528 / (-361.7)) + 180 degrees or arcTan(528 / (-361.7)) + pi radians. This would give you about 121 degrees, which is about 59 degrees north of west.
*@
.
.
.
@&
You're on the right track with a lot of these problems.
I've inserted a number of notes, and if you have additional questions I'm glad to answer.
*@