ANGULAR DIFFERENCE

SUBJECT: MATHEMATICS

CLASS: SS 3

TERM: FIRST TERM

WEEK 8                              Date: ……………………

Topic

ANGULAR DIFFERENCE

The angle subtended at the centre of the great or small circle by the minor arc joining two places on the great or small circle respectively, is called the angular difference between the two places. We shall consider the following:

1. differences in angles of longitude
2. differences in angles of latitude.

Differences in the angles of longitude

When the two points are located on the same side on the earth’s surface i.e. Either East-East or West –West, subtract the angles, otherwise, add them up.

Examples

1. Abuja is located on 32^o E while Lagos is   on  15^oE. What is the difference in their angles of longitude?

(both lie on Lat. 60^oN).

1. Calabar is on 15^oE of the meridian , while Lokoja is located on 12⁰N, find the difference in their angles of longitude (both on latitude 40^o S)

Solution

Difference in angle   = 32^o – 15^o = 17^o

Angular difference  = 12 + 15 = 27^o

Evaluation

1. Ogun is located on 10⁰W and Uyo on 50^o W. Both lies on latitude 40^oN. What is the angular difference?
2. Find the angular difference between the following pairs of places on the Earth’s surface.

• Ketu ( 32^oS, 20^oE) and Lekki ( 32^oS, 65^oE)
• Gombe (20^oN, 10^oW) and Abuja  (20^oN, 45^oE)

Differences in the angles of latitude

When two points are located on the same side along the same longitude on the earth’s surface i.e North-North, South-South, subtract the angles otherwise add them up.

Example:  Find the angular difference between the following pairs of places on the Earth’s surface.

• Paris (35^oN, 40^oE) and Rome ( 25^oS, 40^oE)
• Oyo ( 80^oS, 27^oW) and Edo (47^o S, 27^oW).

Solution

Angular difference = 25 + 35 = 60^o

Angular difference = 80 -47 = 33^o

Evaluation

Find the angular difference between the following pairs of places on the Earth’s surface.

(a) P (30^oS, 48^oE)      and Q  ( 60^oS, 48^oE)

(b) R ( 22^oS, 60^oW)  and S (38^oN and 60^oW)

DISTANCES ALONG THE GREAT CIRCLES

The great circles are only lines of longitude and equator which is 0^o.  Therefore distances along the lines of longitude will majorly be the focus.  Their radii are the same as the radius of the earth ( R).  The knowledge of mensuration shall be used  to calculate  the distance along the lines of longitude.

Distance along the great circle   = Ө^o   x 2πR  (where R is the radius of the earth)

360⁰

Example 1

The position of Libreville (Gabon and Kampala (Uganda) to the nearest degree are (0^oN, 90E) and (0^oN,32⁰E) respectively.  Calculate their distance apart along the equator.

Solution

Step 1:  Sketch a simple diagram of the earth and locate the position of Libreville and Kampala using the values given. (note that liens of latitude are positioned either closer to the North pole or South pole while the lines of longitude are positioned either to East or West of the Meridian.

The distance required is LK

Step 2:  Find the difference between the angles of the longitude  .  Ө^o = <EOK – < EOL

:. Ө^o = 32 – 9^o  = 23^o

Step 3: Calculate the distance  LK ( are LK) using the mensuration formula to calculate the length of arc of a circle.

:. LK  = Ө^o   x    2πR  (where R is the radius of the earth)

360⁰

=  23^o  x 2   x 22/7  x 6400 km

360^o  = 25.68km

= 2600km to 2 s.f.

Example 2:

An aeroplane from a town P (lat. 40^oN, 38^oE) ^o another town Q (lat.40^oN,22^oW) it later flies to a third town T  (28^oN,22^oW) . Calculate the distance between Q and T. along the lines of longitude.

Solution

Step 1:  Sketch a simple diagram showing the position of the points on the earth’s surface.

Step 2:  Since Q and T lie on the same longitude (22^oN) but different latitude ,calculate the difference in their angles of latitude using the same method as earlier  done above i.e if the two latitude lie on North or South, subtract the angles, otherwise add them together .

Ө = <EOT  = < EOQ

:. Diff = 40^o – 28^o = 12^o         Ө = 12^o

Step 3:  Calculate the distance QT along the line of longitude

QT = Ө^o  x 2 πR

360^o

= 1341 km  = 1300km to s.f.

Example 3:

Chicago (USA) and Tokyo (Japan) both lie on longitude 25.9^oE. their latitudes are 31.6^oN and 24.8^oS respectively  Calculate the shortest distance between the towns.

From the figure above. <TOC  = < TOE  + <COE

<TOC = 31.6^o + 24.8^o  = 56.

arc BG  = 56.4   x 2 π R

360

= 56.4  x 2 x 22/7 x 6400

360

=  6270km to 3s.f.

Evaluation

The position of Abuja (Nigeria ) and Bonn (Germany) to the nearest degree are (9⁰N, 7⁰E) and (51⁰N,7⁰E) respectively. Use R= 6400km, to calculate their distance apart to 2.s.f.

DISTANCES ALONG PARALLELS OF LATITUDE

Radius of a parallel of latitude .

Consider the diagram sketched below:

Assuming  points P and E are located on the longitude due East of the meridians as shown above.  If O is the centre of the earth and C is the centre of the latitude of P.  The angle  of latitude of point P is θo.  If r is the radius of the parallel of latitude through P, then on     PCO.

CPO  = θ (alternate <s CP//OE)

OP    = R (Radius of the Earth)

Cos α = ^r/_Rr = R Cos α

The above relation is used to calculate the radius of any parallel l of latitude except equator.

Example1.

Find the distance measured along the parallel of latitude, between two points whose latitudes are both 56^oN and whose longitudes are 23^oE and 17^oW respectively.

Solution

Sketch the diagram and locate the two points

Step 2: Since the distance needed is along the parallel of latitude, calculate the  radius of the latitude

r = R Cos α  ( where α  is the angle of the latitude)

r= 6400 x Cos 56^o

Step 3: Calculate the difference in the angles of the longitudes.

α = 23⁰  + 17^o  = 40^o

Step 4:  Calculate the distance AB

arc = AB   = Ө   x 2πr where r = R Cos α

360

= Ө   x 2πRCos α

360

=  40   x 2 x ²²/7  x 6400 x Cos 56^o

360

=  40,000 x 0.5592

9

=    2490km

Example 2

Two points M and N on the surface of the Earth are given by their latitudes and longitudes as M ( 50^oS, 15^oE) and N(50^oS, 75^oE) calculate :

1. the radius of the parallel of latitude on which M and N lie
2. the distance MN measured along the parallel of latitude (take R = 6400km)

Solution

Step 1: Locate point M and N on a simple diagram

Step 2: Calculate the radius of latitude of the two points.

r = R Cos θ

r  = 6400km x Cos 50

Step 3:  Go ahead and calculate the distance MN but firstly, we need to know the difference in the angles of longitude.

:.θ= 75^o – 15^o  = 60^o

:. MN  = θ       x 2πr

360

= 60^o x 2 x ²²/₇  x 6400 Cos 50

360

=  4740km  , = 4740km to 2. sig.fig.

Example 3

R (lat 60^oN, long 50^oW) is a point on the earth’s surface. L is another point due east of K and the third point N is due south of K.  the distance KL is 3520km and KN is 10951KM. Calculate

1. a) the longitude of L b) the latitude of N. (take π = 22/7 and R = 6400Km)

Solution

Step 1: Sketch the diagram and try to locate the points.

Step 2: Calculate the radius of latitude K and L

R= R Cos Ө

r = 6400 x Cos 60^o , r

Step 3: Find the difference in the two angles of longitudes K and L

K = 50^oN  L is due east of K (Ө^o)

:.  Diff = 50^o + Ө

Step 4: since KL is 3520km, find the value of Ө^o

KL = Ө^o + 50^o  x 2 x 22/7  x 6400 Cos 60^o

360^o

3520 = 50 + Ө^o x 22 x 6400

360 x 7

:.  50 + Ө^o  = 63^o

Ө^o  = 63 – 50^o

:. Ө^o  = 13^o

<;L is 13^oE.

To find the latitude of N:

Step 1: Find the difference between the angles of latitudes  K and N.

K = 60^oN, N is due South of K

:. Diff = 60^o + Ө^o

Step 2:  Since the distance is along the great circle , there is no need to calculate the radius of the latitude.

KN = 60^o + Ө^o  x 2 πR

360^o

10951  = 60^o + Ө^ox 2 x 22/7 x 6400

360

60⁰ + Ө^o  = 10951 x 2 x 22/7’ x 6400

44 x 6400

60⁰ + Ө^o  = 98^o

Ө^o = 98^o – 60^o

:. The latitude of N = 38^oS.

Evaluation

1.A plane flies due East from A (lat. 53⁰N, long 25⁰E) to a point B (lat 53⁰N, long. 85⁰E) at an average speed of

400km/hr. the plane then flies due south from B to a point C, 2000km away. Calculate correct to the nearest whole number:

1. the distance between A and Bb. the time the plane takes to read point B c . the latitude of C

(Take R = 6400km and π= 22/7)

Reading Assignment

New General Mathematics  Chapter 7 pg 58-61,

Essential Mathematics for SS3, by AJS Oluwasanmi  chapter 8 pgs 104-107

General Evaluation

1. Find  the  distance  between  Bida  (9⁰N,  6⁰E)  and  Addis  Ababa  (9⁰N, 38.8⁰E)  measured  along  the  parallel  of  latitude.
2. Two  places  P and  Q  with  longitudes  of   152⁰E  and  171⁰W both  lie  on  the  Equator.Find  the  shortest  distance  between  P  and  Q  on  the  Earth s  surface.
3. Two  places  P  and  Q  have  the  same  longitude. Their  latitudes  are  150N  and  360S.Calculate  the  great  circle  distance  between  PQ.
4. Two  places  X  and  Y  on  the  equator  are  on   longitude  670E  and  1230E  respectively.

(a)What  is  the  distance  between  them  along  the  equator.

(b)How  far  is  X  from  the  North  pole?(Take  π = 22/7  and  the  radius  of  the  earth  as  6400km)

Reading Assignment:

NGM Chapter 7 pg 53-57

Essential Mathematics by AS. Oluwasanmi

                                                                    Weekend Assignment

1. The radius of Great Circles is approximately ……(a) 6500km (b) 6400km  (c) 6500m    (d) 6400m
2. The circumference of a great circle with a radius of 6400km is approximately ……(a) 6200km  (b) 6400km

(c) 6300km         9d) 6100km

3. P (44⁰N, 10⁰E) and Q (36⁰S, 10⁰E), what is the angular difference between point P and Q on the Earth’s surface?(a) 8^o (b) 80^o (c ) 70^o (d) 20^o
4. What is the radius of parallel of latitude 30⁰N?(a) 5530Km (b) 5540Km      (c)  5440m      (d) 5340Km
5. What is the circumference of latitude 30^o N ? (a) 34800km (b) 34900km    (c) 35800km   (d) 34700km

Theory

1. Alakuko and Meiran lie on the equator on longitude 7⁰E and 90⁰W respectively, if the circumference of the earth is 4000km find their distance apart to 3s.f.

1. The position of A is 38^oN, 73^oE and the position of B is 38^oN, 107^oE. Calculate the distance from A to B

(a) along the parallel of latitude

(b ) along the great circle.