# SS 3 MATHS NOTE WEEK 1

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1 SS 3 MATHS NOTE WEEK 1 Longitude and Latitude Lines of latitude run East & West or horizontal but measure distance North & South of the Equator vertically. The equator is labeled as zero degrees latitude. The greatest amount of latitude is 90 degrees at the North or South poles. We can label our equator as 0 degree. Latitude are parallel to the equator and are even referred to as parallels or parallels of latitude. Moving to lines of longitude, they run perpendicular to lines of latitude. That is, longitude lines run North and South but measure East and West of zero degrees longitude which is a line called the Prime Meridian. This arbitrary north/south line was marked by the British in the 17th century and runs through a town just outside of London called Greenwich. Starting at the Prime Meridian, we measure the earth east or west with these north/south-running lines called meridians. We can measure halfway around the world till these meridians meet at 180 degrees. This meridian line at east or west is called the International Date Line. So unlike latitude, 180 at either the north or south where the greatest or maximum latitude is 90 halfway around the world from poles, the greatest amount of longitude is 180 the prime meridian. One other important way these longitude lines differ from parallel latitude lines is that lines of longitude are not parallel, and in fact converge at both the North and South poles. CLASS ACTIVITY: Teacher Write on a board or a sheet of chart paper the terms latitude and longitude. Let students share what they know about the terms. Write down information that students share. Correct any misconceptions they might share.

2 Explain that the lines of latitude and longitude (see sidebar for tricks to help students remember the difference) comprise an imaginary grid that has been placed over the globe. The lines that run across the grid -- the flat lines -- are lines of latitude. The Equator is an example of a line of latitude. The Equator is the latitude line that divides Earth into two hemispheres, the northern hemisphere and the southern hemisphere. The Equator is the 0 point of latitude. Latitude lines North of the Equator are referred to as North latitude; latitude lines South of the Equator are referred to as South latitude. The lines that run up and down on the grid -- the tall lines -- are lines of longitude. The lines of longitude are also called meridians. The Prime Meridian, which passes through Greenwich, England, is the longitude line that divides Earth into two hemispheres, the eastern hemisphere and the western hemisphere. The Prime Meridian is the 0 point of longitude. Longitude lines East of the Prime Meridian are referred to as East longitude; longitude lines West of the Prime Meridian are referred to as West longitude.

3 MEANING OF LATITUDE & LONGITUDE Latitude and longitude are measuring lines used for locating places on the surface of the Earth. They are angular measurements, expressed as degrees of a circle. A full circle contains 360. Each degree can be divided into 60 minutes, and each minute is divided into 60 seconds. The symbol for minutes is ( ) and for seconds is ("). Latitude lines (called parallels) circle the planet's surface in lines parallel to the equator, which lies halfway between the North and South poles. Latitude lines circle the planet from east and west, beginning at 0 at the equator and increasing to 90 North or 90 South. Longitude lines (called meridians) run north and south and circle the planet from pole to pole. Each line is identified by the number of degrees east or west from a starting point at Greenwich, England, known as the prime meridian. The longitude at the prime meridian is 0. Longitudes lines mark the distance westward, to 180 West, and eastward, to 180 East. Latitude and longitude measurements can be written in several ways. For example, 211 degrees, 22 minutes, and 30 seconds is written as ", or with spaces: ; or with a decimal after the degrees ", or with a decimal after the minutes One degree of latitude equals about 69 miles. One minute is just over a mile, and one second is about 100 feet. POINTS TO NOTE: 1.1. Rotation of the Earth The spinning of the earth on its imaginary axis is called rotation. Aside from the cultural influences of rotation, this spinning also has a physical influence. The spinning has led to the creation of a system to determine points and directions on the sphere.

4 The North and South poles represent the axis of spin and are fixed reference points. If the North Pole was extended, it would point to a fixed star, the North Star (Polaris). Any point on the earth's surface moves with the rotation and traces an imaginary curved line: Parallel of Latitude 1.2. The Equator If a plane bisected the earth midway between the axis of rotation and perpendicular to it, the intersection with the surface would form a circle. This unique circle is the equator. The equator is a fundamental reference line for measuring the position of points around the globe. The equator and the poles are the most important parts of the earth's coordinate system Parallels of Latitude Parallels of latitude are all small circles, except for the equator. True east-west lines Always parallel Any two are always equal distances apart An infinite number can be created Parallels are related to the horizontal x-axes of the cartesian coordinate system. Figure 4. Parallels of Latitude Meridians of Longitude Meridians of longitude are halves of great circles, connecting one pole to the other. All run in a true north-south direction Spaced farthest apart at the equator and converge to a point at the poles An infinite number can be created on a globe Meridians are similar to the vertical y-axes of the cartesian coordinate system Degrees, Minutes, and Seconds Angular measurement must be used in addition to simple plane geometry to specify location on the earth's surface.

5 This is based on a sexagesimal scale: A circle has 360 degrees, 60 minutes per degree, and 60 seconds per minute. There are 3,600 seconds per degree. Example: 45 33' 22" (45 degrees, 33 minutes, 22 seconds). It is often necessary to convert this conventional angular measurement into decimal degrees. To convert 45 33' 22", first multiply 33 minutes by 60, which equals 1,980 seconds. Next add 22 seconds to 1,980: 2,002 total seconds. Now compute the ratio: 2,002/3,600 = Adding this to 45 degrees, the answer is The earth rotates on its axis once every 24 hours, therefore: Any point moves through 360 a day, or 15 per hour Great and Small Circles A great circle is a circle formed by passing a plane though the exact center of a sphere. The largest circle that can be drawn on a sphere's surface. An infinite number of great circles can be drawn on a sphere. Great circles are used in the calculation of distance between two points on a sphere. A small circle is produced by passing a plane through any part of the sphere other than the center. 1.4 Latitude Authalic Latitude is based on a spherical earth: Measures the position of a point on the earth's surface in terms of the angular distance between the equator and the poles. Indicates how far north or south of the equator a particular point is situated. North latitude: all points north of the equator in the northern hemisphere South latitude: all points south of the equator in the southern hemisphere Latitude is measured in angular degrees from 0 at the equator to 90 at either of the poles. A point in the northern hemisphere 40 degrees north of the equator is labeled Lat. 40 N. Forty degrees south of the equator, the label changes to Lat. 40 S.

6 The north or south measurement of latitude is actually measured along the meridian which passes through that location It is known as an arc of the meridian Longitude Longitude measures the position of a point on the earth's surface east or west from a specific meridian, the prime meridian. The longitude of a place is the arc, measured in degrees along a parallel of latitude from the prime meridian. The most widely accepted prime meridian is based on the Bureau International de l'heure (BIH) Zero Meridian: Defined by the longitudes of many BIH stations around the world.. Passes through the old Royal Observatory in Greenwich, England. The prime meridian has the angular designation of 0 longitude. All other points are measured with respect to their position east or west of this meridian. Longitude ranges from 0 to 180, either east or west. Since the placement of a prime meridian is arbitrary, other countries have often used their own. For the purposes of measurement, no one prime meridian is better than another Having a widely accepted meridian allows comparison between maps published in different areas. The distance represented by a degree of longitude varies upon where it is measured. The length of a degree of longitude along a meridian is not constant because of polar flattening. At the equator, the approximate length is determined by dividing the earth's circumference (24,900 miles) by 360 degrees: kilometers (69 miles). The meridians converge at the poles, and the distance represented by one degree decreases. At 60 N latitude, one degree of longitude is equal to about kilometers (34.5 miles).

7 WEEK 2 DISTANCE ALONG PARALLELS OF LATITUDE: (radius of parallel of latitude) N C r P R O R E S Let r be the radius of the parallel of latitude through P, then In Δ PCO, CPO=θ OP =R Cos θ= r=rcosθ Example Find the distance, measured along the parallel of la tude, between two points whose la tude are both 56 N and whose la tudes are respec vely 23 E and 17 W. Solution ACB= = 40 C r B A Arc AB= θ ₓ 2πr 360 NB: Since r is radius of parallel of latitude, then r=rcosθ and 2πR= θ= 40 π= 22 7 Arc AB= θ ₓ 2πr 360

8 = 40 ₓ 2πR Cosθ 360 = 40 ₓ Cos = km = 2490km (3 s.f) Example 2 An aeroplane leaves a point P(21 N, 32 E) and flies due south at 850 km/h. Find its position Q after 4 hrs. (Assume that the earth is a sphere of radius 6400km). Solution P 6400km O θ Q Let POQ=θ Θ=21 + x PQ=850 ₓ 4 = 3400km Arc= PQ= θ ₓ 2πr = 40 ₓ 2 ₓ 22 ₓ = 7040θ 63 Θ= = 30 (2 s.f) The posi on of Q = x = = 9 S NB: Pls do yourself good by carrying out the activities in the project work and also attempt solving the questions given to you.

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