MAC 1147 Broward College Trigonometry Equations on Interval Problems show work for all questionsthis is for my MAC1147 Spring 2020 Final Project only answe

MAC 1147 Broward College Trigonometry Equations on Interval Problems show work for all questionsthis is for my MAC1147 Spring 2020 Final Project only answer problems 1-6. do not do extra credit MAC1147 Spring 2020 Project
Show all steps for full credit
Problem 1: Graphing Trigonometric functions (Difficulty: Easy) (20 points)
Graph two full periods of the functions below. Determine the Amplitude, Period, Quarter Period, Phase
Shift and a table of points in order to receive full credit
??
?? = ?2 sin (2?? + )
2
1
?? = cos(3?? ? ??)
2
??
?? = 3 tan (?? + )
4
??
?? = 2 cot (?? + ) ? 1
6
?? = csc(?? ? ??)
??
?? = sec (2?? + ) ? 1
2
Problem 2: Model real life data using Trigonometry (Difficulty: HARD) – 100
points
In this problem, you will collect 10 years of real-life weather data and fit a trigonometric equation to
model and predict future temperatures. Visit NOAA’s Global Climatology Network which is an
aggregator of daily temperatures in many places around the world. When you first visit, it’ll take you
through a lengthy tutorial. Simply go through it. In this walkthrough, we will extract Daily Maximum
Temperature for Miami. If you would rather do this exercise for your country/city of birth, or rather a
different city than Miami, then you’re welcomed to do so.
We want to extract 10 years of
Type “Miami”. Click on the
Weather Data. Select the dates
Type “Maximum
pop-up blue block to finalize
as seen below. Make sure
Temperature” Click on the
your choice
“Select Range” is selected
pop-up blue box to finalize
your choice
Type “Miami Beach, FL
US” and click on the
pop-up blue box to
finalize your choice.
NOTE: DO NOT PICK
MIAMI BEACH 2, FL US
If done right, you will
see something similar.
File Count/Limit:
1/1000 and 1 Station
Miami Beach, FL US
Click on Select All to
make a selection
Once you click on “Select All”, you should see a pop-up that looks like this. Click on “Configure and
Add”. Choose CSV (Excel/Google Sheet format) or PDF
Next Page, click on “Add Order To Cart”
Review your order. Yours should look like this.
Once satisfied, put your
email. Weather data will
be delivered by email
You will receive two emails: One about the order you submitted and another with the actual data. It
may take up to 5minutes to receive the email with the actual data. Check your Junk Box if you don’t see
it after 5mns.
If the Data format chosen was CSV, you can open the data in Excel or Google Sheets. It should look like
the picture on the left. On the Right is a PDF version. Know that you’ll need to type everything by hand
when it’s time to analyze the data should you choose PDF version.
Question 1: From the data you gathered, plot a Scatter Plot using Excel/Google Sheet or other software
that can plot raw data. Refer to Online examples if you’re not sure how to use Excel to Plot.
Question 2: Find the Amplitude, Period, Quarter Period and Phase Shift and Vertical Shift of your data,
then construct the Trigonometric model of your data. Plot one period using the equation you obtained.
Extra Credit: For Bonus Points, plot 10 years of data (from Question 1) overlaid with the actual equation
you calculated from Question 2 using excel or any other software that can plot data.
Problem 3: Solve each equation on the interval [??, ????) (20 points)
tan(3??) = ?1
??
?2
sin (2?? ? ) =
4
2
2 cos 2 ?? + 3 cos ?? + 1 = 0
2 cos 2 ?? + sin(??) ? 1 = 0
tan2 ?? cos(??) = tan2 ??
??
??
sin (?? + ) + sin (?? ? ) = 1
3
3
sin(2??) cos(??) + cos(2??) sin(??) =
?2
2
tan(??) + sec(??) = 1
2 sin3 ?? ? sin2 ?? ? 2 sin(??) + 1 = 0
sin(3??) + sin(??) + cos(??) = 0
Problem 4: Unit Circle. (Difficulty Medium) (20 points)
Most unit circles are constructed using the following angles: 0, 30, 45, 60, 90 and their multiples. Your
task is to fill in a very different unit circle. Use any trigonometric tool at your disposal to fill in the unit
circle below. Note all angles are in degrees.
Problem 5: (Difficulty: Easy) (20 points)
The table below gives the spatial location of 8 planets on an X-Y coordinate system. Assume the sun is at
the origin of the Cartesian coordinate system.
Planet
Sun
Mercury
Venus
Earth
Mars
Jupiter
Saturn
Uranus
Neptune
X
0
-157
-200
+268
+1450
+2871
+8543
+19922
+26272
Y
0
+279
+698
+979
+694
+4098
+4635
+1988
+14355
R
??
Question 1: Plot the location of each planet in a Cartesian Coordinate System. Assume spacing to be
1000 units. Label each planet accordingly
Question 2: Plot the location of each planet on a Polar Coordinate System. Assume the first 9 concentric
circles are spaced by 200 units each, the next 3 are spaced 4000 units each and the last 2 at 11,000
units. Label each planet accordingly
Question 3: What is the distance between the Sun and Mars if the angle between the Sun and Mars, as
viewed from Earth, is 88 degrees? Calculate the distance between the Sun and Mars using Law of
Cosine.
Problem 6 (Difficulty: Medium) (20 points)
The Ares 1-X was the first stage prototype and design concept demonstrator in the Ares 1 program, a
launch system for human spaceflight developed by NASA. For more information, research the Ares 1-X
program. Ares 1-X vehicle underwent powered flight. After reaching its apogee (highest point in
trajectory) the vehicle coasted and landed back to Earth. Its parametric equation is defined as follows
??(??) = 64000 + 1800??
??(??) = 45000 ? 4.9?? 2
Where x(t) is the vehicle’s horizontal location and y(t) its altitude in meters and t is time of flight in
minute
Question 1:
Convert the parametric equation to a new polynomial function, f(x), by eliminating the variable t.
Question 2:
Determine height from ground the vehicle reached at its apogee and how far from its launch point at
Cape Canaveral the capsule landed. Distances are in meters
Question 3:
Using the result from question 2, determine how long the flight lasted. Time is in minutes
Apogee
Launch Point
Landing Point
Extra Credit: (Difficulty Medium) (20 points)
In 1990, The Mathematical Intelligencer, a mathematical journal, asked its readers to choose the most
beautiful theorem in Mathematics, and Euler’s (pronounced oiler) famous identity was picked. It is:
?? ????? + 1 = 0
Where:
?? is Euler’s number, the base of natural logarithms (?? = 2.71828 …)
?? is the complex unit, which is defined as ?? 2 = ?1, and
?? is pi, the ratio of circumference of a circle to its diameter
This formula is a special case of the general formula which states:
?? ????? = cos(??) + ?? ? sin (??)
We can easily see the relationship between the general formula and the fore-mentioned identity.
If ?? = ??, then
?? ???? = cos(??) + ?? ? sin(??)
?? ???? = ?1 + ?? ? (0)
?? ???? = ?1
?? ???? + 1 = 0
Question 1: Given that ?? ???? = cos(??) + ????????(??), only use that identity to show: ?? ???? ?? ????? = 1
Many other trigonometric identities can be derived using the general Euler’s formula. We can show the
cosine and sine of the sum of two angles are true.
?? ??(??+ ??) = ?? ???? ?? ????
The left-hand side is:
?? ??(??+??) = cos(?? + ??) + ??sin (?? + ??) (eqn 1)
The right-hand side is:
?? ???? ? ?? ???? = [cos(??) + ????????(??)] ? [cos(??) + ????????(??)] (eqn 2)
If you FOIL the equation (eqn 2) using only the right-hand side, you get:
?? ???? ? ?? ???? = cos(??) ? cos(??) + ?? ? cos(??) ? sin(??) + ????????(??) ? ??????(??) ? sin(??) ? ??????(??) (eqn 3)
From (eqn 3), we can group all the terms with ?? together and terms with no ??. We get:
?? ???? ? ?? ???? = [cos(??) ? cos(??) ? sin(??) ? sin (??)] + ??[cos(??) ? sin(??) + ??????(??) ? ??????(??)] (eqn 4)
Setting the two equations together (eqn1 and eqn4) by equating the terms with/without ??, we obtain:
cos(?? + ??) + ??(sin (?? + ??) = cos(??) ? cos(??) ? sin(??) ? sin (??) + ??[sin(??) cos(??) + sin(??) sin(??)]
cos(?? + ??) = cos(??) ? cos(??) ? sin(??) ? sin (??) and sin(?? + ??) = sin(??) cos(??) + sin (??)sin (??)
Question 2: Using the same technique above, demonstrate the followings
a) Difference of two angles using ?? ??(?????) = ?? ???? ?? ?????
2
b) Double angle formula using ?? 2????? = (?? ???? )
c) Product to Sum formula using ?? ??(??+??) + ?? ??(?????) = ?? ???? ?? ???? + ?? ???? ?? ?????

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