#$&*
Mth 164
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
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Hello! I am working on assignment 2; I have completed the class notes/qa, which I understood just fine. Although, now I am trying to do the Chapter 5 questions, and I cannot figure out where some of the information is coming from. I have looked through my text book, and found a few things, but I still have way to many questions to be able to complete the assignment. Here are the questions I am talking about;
Section 5.2 (6.2 starting with 8th edition)
For the given points on the terminal side of angle `theta, find the exact value of the six trigonometric functions of `theta:
6. ( 1, -1)
@& The point (1, -1) lies on a unique circle centered at the origin. You can find cos(theta) = x/r, etc., where for example x is the x coordinate of this point (i.e., x = 1) and r is the radius of the circle.
Try answering the following questions:
If you sketch a line segment from the origin to (1, -1), what is the length of that segment?
What therefore is the radius of the circle through the origin that passes through (1, -1)?
What therefore are the values of x / r, y / r, y / x, r / x, r / y and x / y? To which trigonometric functions do these ratios correspond?*@
10. (-0.3, -0.4)
Without using a calculator find the exact value of each expression:
12. sin 30 deg - cos 45 deg
18. sec (30 deg) * cot (45 deg)
20. 5 cos( 90 deg) - 8 sin( 270 deg)
24. tan(`pi/3) + cos(`pi/3)
30. sec `pi - csc (`pi/2)
@& I believe the 30-60-90 and 45-45-90 triangles are covered in Section 6.3 of your text; I don't have a copy here and can't verify this, but it's one of the things I checked when doing the correspondences. However I could be wrong about this.
In any case this is in your text and you can look it up if you need more detail than I give you here.
@& This is also covered in Class Notes #04, which however aren't assigned until the next assignment. You can look at that video and/or the document posted on the website, to help you visualize the discussion below.*@
An equilateral triangle with sides 1 has three 60-degree angles. If you draw a line segment from any vertex to the midpoint of the opposite side, it divides the triangle into two right triangles, each containing angles of 30, 60 and 90 degrees. The hypotenuse will be one of the original sides, and its length will be 1. The line you drew will be another, and its length is therefore unknown. The third side will be half of one of the original sides so its length will be 1/2.
Using the Pythagorean Theorem you easily find that the unknown leg has length sqrt(3) / 2.
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@& It follows that the points (+-sqrt(3) / 2, +-1/2) lie on the unit circle, at angles of 30, 150, 210 and 330 degrees; while the point (+-1/2, +- sqrt(3) / 2) lie at 60, 120, 240 and 300 degrees.*@
@& A right triangle with two equal legs, each of length 1, has hypotenuse sqrt(2). It follows that the angles 45, 135, 225 and 315 degrees will intersect a circle of radius sqrt(2) at points of the form (+-1, +-1).
So for example a line at 45 deg relative to the positive x axis intersects a circle of radius sqrt(2) at the point (1, 1), so that cos(45 deg) = x / r = 1 / sqrt(2) = sqrt(2) / 2.
*@
Find the exact values of the six trigonometric functions of:
36. - `pi / 3
40. 3 `pi
42. -270 deg
If `theta = 60 deg find the exact value of:
72. cos(`theta)
78. cos( 2 * `theta)
80. 2 cos(`theta)
84. Find the exact value of tan 60 deg + tan 150 deg.
90. If cos( `theta) = 2/3, find sec (`theta).
@& For any angle that is a multiple of 30 deg, or pi/6 radians, but not of 90 deg or pi/2 radians, we can therefore locate the exact x and y coordinates on a circle of radius 1. One of the coordinates will have magnitude 1/2, the other will have magnitude sqrt(3) / 2. Using these coordinates (and, of course, r = 1), along with + or - signs appropriate to the quadrant, we can find the values of the six trigonometric functions.
If the angle is a multiple of45 deg, or pi / 4 radians, we can locate the corresponding point on a circle of radius sqrt(2). Both x and y coordinates will have magnitude 1.
You will have plenty of practice with this in the next assignment, and if you prefer you can wait until then to answer the questions about exact values.*@
Sec 5.3 (6.3 starting with 8th edition)
Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator.
6. Sec 540`theta deg
10. sin 9 `pi / 4
12. csc 9 `pi /2
@& sin(theta) is defined to be y / r.
cos(theta) is defined to be x / r.
r is always positive.
So for example of cos(theta) < 0 and sin(theta) > 0 it follows that x < 0 and y > 0, putting us in the second quadrant.*@
18. sin `theta < 0, cos `theta > 0
20. cos `theta > 0, tan `theta > 0
24. csc `theta > 0, cos `theta < 0
in the next problem sin `theta and cos `theta are given. Find the exact value of each of the four remaining trigonometric functions.
30. sin `theta = 2`sqrt(2) / 3 , cos `theta = -1/3
find the exact value of each of the remaining trigonometric functions of `theta.
36. sin `theta = -5/13, `theta in quadrant 3
40. sin `theta = -2/3, `pi < `theta < 3`pi/2
42. cos `theta = -1/4, tan `theta > 0
48. sec `theta = -2, tan `theta > 0
use the even- odd properties to find the exact value of each expression. Do not use a calculator.
50. cos (-30 deg )
54. csc (-30 deg)
60. sin (-`pi/3)
66. csc (-`pi/3)
find the exact value of each expression. Do not use a caculator.
@& You can do these by locating these angles on a circle. The angle is always measured from the positive x axis, and the counterclockwise direction is positive.
For example 19 pi / 4 is 4 pi + 3 pi / 4.
4 pi is two complete trips around the circle. So 19 pi / 4 is two complete trips, plus 3 pi / 4.
It follows that 19 pi / 4 occurs at the same point on the circle as 3 pi / 4.
This is a multiple of pi / 4, so on a circle of radius sqrt(2) the magnitudes of both the x and y coordinates are 1. The angle 3 pi / 4 is in the second quadrant, so we have x = -1 and y = 1.
Thus cos(19 pi / 4) is x / r = -1 / sqrt(2) = -sqrt(2) / 2.*@
70. tan (-6`pi) + cos (9`pi/4)
72. cos (-17`pi/4) - sin (-3 `pi/2)
78. cot 20 deg - cos 20 deg/sin20 deg
80. If cos `theta = 0.2, find the value of cos `theta + cos ( `theta+2`pi) + cos (`theta+4`pi).
84. What is the domain of the cosine function?
90. What is the range of the cosine function?
96. If the cosine function even, odd or neither? Is its graph symmetric? With respect to what?
100. Is the cosecant function even, odd, or neither? Is its graph symmetric? With respect to what?
102. If f(x) = cos x and f(a) = 1/4, find the exact values of:
f(-a)
f(a) + f(a+2`pi) + f(a - 2`pi)
110. Use the periodic and even- odd properties to show that the range of the cotangent function is the set of all real numbers.
I looked ahead to the query and there are lots of questions I don't understand there too.
I also looked ahead to the next class notes, and much of the information appears to be there. Should I review those? This is all just confusing me because that would seem to be out of order. Help!
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I do not understand how to find the exact values of the expressions that have sin, cos, tan, etc. in them. Really, I couldn't figure much out about that at all.
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Where do I find the information I need to learn to do this work? Am I missing something? Please help get me back on track!
@& I'll have to check your book on Wednesday afternoon to verify that this information in fact corresponds to the assigned sections of your text in the way I thought it did, and I can give you more detail then. I can also check the phrasing of problems in your text and more closely compare it with the phrasing we see here.
However the notes I've inserted here should be helpful.
You might also look ahead to some of the upcoming class notes, and/or look at Sections 6.1 and 6.2 of your text (which will soon enough be assigned in any case).*@