HSC Mathematics General 2- Trigonometry, Spherical Geometry
and Data Analysis
NAME: _________________________________________
Outcomes Assessed
MG2H-1 uses mathematics and statistics to evaluate and construct arguments in a range of
familiar and unfamiliar contexts.
MG2H-2 analyses representations of data in order to make inferences, predictions
and conclusions.
MG2H-4 analyses two-dimensional and three-dimensional models to solve practical problems,
including those involving spheres and non-right-angled triangles.
MG2H-7 answers questions requiring statistical processes, including the use of the normal
distribution, and the correlation of bivariate data
Topics Assessed
MM5 Applications of trigonometry
MM6 Spherical geometry
DS4 Interpreting sets of data
Part 1: Trigonometry in non-right angled triangles
a) Select 3 points using Google earth which form a NON right angled triangle
REFER TO APPENDIX A
Using Google earth, locate a town (or place) of your choice. This can be anywhere in the
world. Zoom in and find a suitable location for the start of a pigeon race. Choose 2 suitable
stopping points for the pigeons along the way. Ensure the triangle formed by these three
points is NOT a right angled triangle. Pigeon races can be between 100 km and 1000 km.
From Google earth, write down the latitude and longitude of the
Start
__________________________________________
1
Stop 1
_________________________________________
1
Stop 2
__________________________________________
1
NOTE: your latitude and longitude only need to be to the nearest degree, as you have seen
them written in class. Google earth may give you more accurate positions, which are not
required for this assignment.
b) Find the distances and bearings from the start to stop 1 and stop 1 to stop 2.
Using the measuring tool in Google earth, take measurements and complete the following
information below. Make sure all measurements are to the nearest kilometre, and
bearings to the nearest degree.
Start to stop 1: distance:
____________________ bearing ________________
2
Stop 1 to stop 2: distance:
____________________ bearing ________________
2
c) Attach the print out of your selected location.
1
d) Use a protractor to find the angle  start/stop1/stop2. Mark this clearly on your print out.
1
e) Use the formulae below and your answers from parts a and b to complete the following
questions.
Area of a triangle =
Cosine rule:
Sine rule:
or
3
Use the measurements you took earlier and an appropriate trigonometric rule to calculate
the direct distance between the Start and Stop 2 points of the race.
State the rules used and show all working, including any diagram.
f) Use the relevant measurements you found in part (b) and an appropriate trigonometric rule
to find the angles
(i)  stop 1/Start/Stop 2
(ii)  stop 1/stop2/start.
Show all working, including any diagrams.
4
g) Calculate the area of the triangle that the pigeons would fly around, showing all working.
2
h) Comment on the accuracy of your answers to the above questions and any factors that may
have affected that accuracy.
2
Part 2 – Spherical Geometry
Refer to appendix B
a) Find the positions of two places with the same longitude, one in the southern hemisphere
and one in the northern hemisphere.
Make sure you include the name of the place in the space provided. Again, your answers
should be to the nearest degree, as you have seen in class.
Place A: ____________________________ (____________________, __________________)
3
Place B: ____________________________ (____________________, __________________)
3
b) Show your working to find the distance between these two places to the nearest kilometre.
(Use the radius of the earth to be 6400 km)
3
c) Find the positions of two places with the same longitude, both in either the northern or
southern hemisphere.
Place C:____________________________ (____________________, __________________)
3
Place D:____________________________ (____________________, __________________)
3
d) Show your working to find the distance between these two places and then convert this
distance to the nearest nautical mile, given the conversion rate that 1 nautical mile (M) =
1.852 km.
4
e) Find two places in the southern hemisphere which have the same latitude. Write down the
position of each place:
Place E: ____________________________ (___________________, _________________)
3
Place F: ____________________________ (___________________, _________________)
3
f) Parallels of latitude are small circles, except for the Equator, which is a great circle.
2
To find the radius (r) of a small circle, we use r = R x cos , where  is the latitude, and
R = 6400 km, the radius of the earth. Find the radius of the parallel of latitude for Places E
and F from (e).
g) Show your working to find the distance between these two places, (nearest kilometre)
3
h) Show your working to find the time difference between these two places.
2
i) If it is 9am on Monday in Place E, what is the time and day in Place F?
2
Part 3: Comparing sets of data
You have been assigned a weather station. See Appendix C for instructions.
a) Write down the number, name of your weather station, as well as its latitude, longitude and
altitude (height above sea level) and year the weather station opened.
Number: __________________________ Name: ______________________________________
Lat and long _____________________________________________
elevation: __________________________ Year opened _____________________________
Find the monthly maximum temperatures for 2013, and another year at least 20 years ago from
that particular area, and write them in the space provided.
2013
JAN
FEB
MAR
APR
MAY
JUN
JUL
AUG
SEPT
OCT
NOV
DEC
YEAR: _______________
JAN
FEB
MAR APR
MAY
JUN
JUL
AUG
SEPT
OCT
NOV
DEC
b) Put each of these sets of scores in order and find the five number summary for each year
2
2
2012
YEAR: _________
c) Five number summaries:
Year ___________________
Lowest=
Q1=
Median =
Q3 =
Highest =
Year ___________________
Lowest =
Q1 =
Median =
Q3 =
Highest =
10
d) Complete a box and whisker plot for each set of data in the space provided below, using the
same scale.
3
e) Compare and contrast the two data sets by examining the shape and skewness of the
distributions, and the measures of location and spread.
f) Do you agree with the statement “the temperature has increased over the past 20 years”?
Explain your answer with respect to the information you have found.
2
APPENDIX A
 You may need to make sure your task bar is unlocked and auto hide selected as you may
need the entire screen. Do this from the control panel accessed from the start menu.
 Open Google Earth (in DirectX mode if you get an option) and zoom in to your desired
location. It can be anywhere. Make sure you rotate your view so North is at the top of
the screen.
 Insert a place marker on your Start position. To do this, choose the place marker ‘pin’
from the menu, and name it. It will place itself in the centre of the screen. You may need
to shrink the place marker pop up screen and shift it so you can see your marker (and the
OK button on the screen). To move your marker to the exact location you want, click and
drag it when the yellow cross hairs are highlighting it. Click on the OK button to fix this
position. If at a later stage you need to move your place marker, right click and select
properties to activate the yellow cross hairs. Make a note of the latitude and longitude of
your Start position. Google Earth gives these in degrees, minutes and seconds. You will
need to round your positions just to the nearest degree (or, in Google Earth, go to Tools,
options, and set the ‘show lat/long’ as dec degrees)
 Choose the locations of Stop 1 and Stop 2. Insert place markers for these locations, and
make note of their positions.
 Now, use the ruler tool to find the distance and bearing from Start to Stop 1. Select the
ruler tool and choose the line option and choose kilometres. Google Earth calls the
bearing a heading, but it means the same thing. Click first on Start and then on Stop 1. A
line will be drawn from Start to Stop 1. Make sure you do these in the right direction, or
your bearings will be incorrect. Also, make sure that you are centring the white cross
hairs over the pin point, otherwise your triangle sides won’t meet at the vertices. Record
your distance (to the nearest kilometre) and the bearing (to the nearest degree). Choose
to save this; another screen will pop up. Call the path Start-Stop1 and save. The line will
then stay. Repeat this step for Stop 1 to Stop 2.
 Select the printer and print this image. You may need to save this (as a PDF onto a thumb
drive) and then print it. Include this in your assignment- marks are given for this, and
you need to use this print out to measure an angle using a protractor.
APPENDIX B:
Google Earth could be used to find these places, but it is quite difficult. It is easier to use
www.findlatitudenandlongitude.com to find these places.
You only need to click on the map; the latitude and longitude for your selected location,
and what is there will be shown in a box. You only need to have the latitude and
longitude to the nearest degree, and the place name may only be a region, or even a
road! Then click on another place; the position of the ‘mouse’ is displayed in a box. Just
keep moving the ‘mouse’ until its latitude or longitude (whichever one you are looking to
make the same) is the same as the first position. When you click on this new position,
the details of your selected location will change to tell you what is there and its latitude
and longitude.
Do not have the places too near each other; in different countries or continents is a good
idea as it gives you a good angular distance for the working you need to do.
APPENDIX C:
 Go to the Board of Meteorology website at www.bom.gov.au
 Go down to ‘Our Services’, and choose Climate and Past weather
In “weather and climate data” choose “weather and climate data”
1. make sure your data is about TEMPERATURE, monthly, highest temperature
2. To get the data, enter the station number that has been assigned to you OR the
station name and then GET DATA.
If there is not 12 months data for 2013 state this on your answer sheet and choose
another year.
If there is not data from at least 20 years ago state this on your answer sheet and use
the earliest year there.
If there are any issues with the data from your station which prevent you from
answering any questions, see your teacher immediately.