Please explain and show all of your work For number 2 please use any address and zipcode for Chicago or a suburb of Chicago

Please explain and show all of your work For number 2 please use any address and zipcode for Chicago or a suburb of Chicago. Please show the address and zipcode as well.

Consider the graph of y = tan x.

The answers are next to the question, but read the longer explanation below.

(a) How does it show that the tangent of 90 degrees is undefined? There is a vertical asymptote at x = 90 degrees (a vertical line that the graph approaches, but never intersects).
(b) What are other undefined x values? 90 degrees (pi/2 radians) plus multiples of 180 degrees.
(c) What is the value of the tangent of angles that are close to 90 degrees (say 89.9 degrees and 90.01 degrees)? They are very large positive numbers, or very small negative numbers (you can find the exact values of tan(89.9) and tan(90.01) on your calculator if you'd like, but the closer you get to 90, the larger or smaller the number will become).
(d) How does the graph show this? The graph increases steeply (y takes on larger and larger values) as it gets closer to 90 degrees from the left, and decreases steeply (y takes on smaller and smaller negative values) as it gets closer to 90 degrees from the right.

The graph of y = tanx looks like this:

The graph is in green, and the vertical lines (representing the asymptotes) are at pi/2 radians, 3pi/2, 5pi/2, 7pi/2, etc. This is the same as pi/2 + multiples of pi (90 degrees + multiples of 180 degrees).

Let's look at the first vertical line to the right of the y-axis. The graph increases steeply as it approaches the line from the left and decreases steeply as it approaches the line from the right. This means the closer you get to 90 degrees from the left (x = 89, 89.9, 89.99, etc.) the larger y = tanx will be. And, the closer you get to 90 degrees from the right (x = 91, 90.1, 90.001, etc.) the smaller y = tanx will be. However, at 90 degrees, tanx does not exist. (If you've learned about limits, the limit at 90 degrees doesn't even exist. the graph approaches infinity as x approaches 90 degrees form the left and it approaches negative infinity as x approaches 90 degrees from the right).

You can also think of this in a non-graphical way: Remember that tanx is defined as sinx/cosx and that you can't divide anything by 0. When is cosx = 0?

2. A nautical mile depends on latitude. It is defined as length of a minute of arc of the earth's radius. The formula is N(P) = 6066 - 31 cos 2P, where P is the latitude in degrees.

(a) Using the Library and other course resources, find the exact latitude (to 4 decimal places) of where you live, used to live, work, or used to work (include the zip code).

Address: 1 S. State St.
Chicago, IL 60603

Latitude: 41.88 (41°52') http://world.maporama.com/idl/maporama/
41.881943 http://www.gorissen.info/Pierre/maps/ googleMapLocationv3.php

So, the latitude is 41.8819 degrees.

(b) Using the latitude found in part a and the formula N(P), find the length of a nautical mile to the nearest foot at that location.

N(P) = 6066 - 31cos(2P)

N(41.8819) = 6066 - 31cos(2*41.8819)

N(41.8819) = 6066 - 31cos(83.7638)

N(41.8819) = 6066 - 31(0.1086)

N(41.8819) = 6066 - 3.3675

N(41.8819) = 6062. 6325

(c) Next, use the formula N(P) to find the latitude where the nautical mile is 6051 feet.

N(P) = 6066 - 31cos(2P)

6051 = 6066 - 31cos(2P)

-15 = - 31cos(2P)

0.4839 = cos(2P)

arcos(0.4839) = 2P

±61.0615 = 2P

P = ±30.5307

(d) Name two cities in the Northern Hemisphere and two in the Southern that are close to the latitude found in part c.

I went to mapquest.com and entered 30.53 in as a latitude and guessed on the longitude.

North (near latitude 30.53): Cairo (near longitude 30), New Orleans (near longitude -90), Enshi, China (near longitude 110)

South(near latitude -30.53): Colesburg, South Africa (near longitude 25), Sao Gabriel, Brazil (near longitude -54), Cook, Australia (near longitude 130)

3. When graphed using polar coordinates, the center of a regular nonagon is at the origin and one vertex is at (6, 0 degrees) or (6, 0 radians). Find the polar coordinates of the other vertices in both degrees and radians.

A nonagon has 9 sides. Each vertex would be 1/9th of the way around a circle of 360 degrees, 2 pi radians (because it is a regular polygon, the vertices would be equally spaced).

In degrees, there are vertices at multiples of 40 degrees:

360/9 = 40 degrees.

0, 40, 80, 120, 160, 200, 240, 280, 320

In radians, there are vertices at multiples of 2π/9.

0, 2π/9, 4π/9, 6π/9 = 2π/3, 8π/9, 10π/9, 12π/9 = 4π/3, 14π/9, 16π/9

The other part of a polar coordinate is the radius. The radius for the first point is given in the problem - 6. That means that, since this is a regular polygon, the radii are the same for all the other vertices.

So, in degrees, the vertices are: (6, 0) (6, 40) (6, 120) etc.

And, in radians, the vertices are: (6, 0) (6 2π/9) (6, 4π/9) etc.