Mathematics Teachers Enrichment Program MTEP 2012 Trigonometry and Bearings Trigonometry in Right Triangles A In right ABC, AC is called the hypotenuse. The vertices are labelled using capital letters. The sides opposite the vertices are labelled with the corresponding lower case letter. In ABC, AB = c, BC = a, and AC = b. Solving Right Triangles c B a b C When asked to solve a right triangle, you are expected to find all missing angles and all missing side lengths. Sometimes you will only be asked to solve for specific missing information. If you are given two sides in a right triangle, you do not require trigonometry to find the third side. You can use Pythagoras Theorem. In the above triangle, a 2 + c 2 = b 2. If you are given two angles in a right triangle or any triangle for that matter, you do not require trigonometry to find the third angle. You can use the fact that the angles in a triangle add to 180. In the above triangle, ABC ˆ + BAC ˆ + ACB ˆ = ˆB +  + Ĉ = 180. We can use basic trigonometry in a right triangle to find other missing information. In the above triangle: sin A = side opposite  hypotenuse = a b sin C = side opposite Ĉ hypotenuse = c b cos A = side adjacent to  hypotenuse = c b cos C = side adjacent to Ĉ hypotenuse = a b tan A = side opposite  side adjacent to  = a c tan C = side opposite Ĉ side adjacent to Ĉ = c a We can use basic trigonometry in a right triangle to find a missing side provided that we know one other side and an angle other than the right angle. We can find a missing angle in a right triangle provided we know two of the side lengths. Two Definitions The angle of elevation is the angle above the horizontal that an observer must up look to see an object that is higher than the observer. The angle of depression is the angle below the horizontal that an observer must look down to see an object that is lower than the observer. Look up to an object Angle of Elevation Horizontal Horizontal Look down to an object Angle of Depression
Example 1 The picture to the right shows Arch 22 in Banjul. An observer positions himself in front of the arch, 45 m directly in front. The angle of elevation to the top of the arch from this point is 38. Determine the height of the arch to the nearest metre. Example 2 A 5 m ladder rests against a building so that the foot of the ladder is 3.5 m out from the bottom of the wall. Determine, correct to the nearest degree, the angle that the foot of the ladder makes with the ground.
Example 3 Two observers Sona and Yoro, 20 m apart observe a kite in the same vertical plane and from the same side of the kite. The angles of elevation from Sona and Yoro are 25 and 55 respectively. Determine the height of the kite to the nearest metre. Solving Non-Right Triangles An acute triangle is a triangle in which all angles are less than 90. An obtuse triangle is a triangle in which one angle is more than 90. When asked to solve a non-right triangle, you are expected to find all missing angles and all missing side lengths. Sometimes you will only be asked to solve for specific missing information. The Cosine Rule and Sine Rule are used to find unknowns in acute and obtuse triangles.
The Cosine Rule The Cosine Rule is used to find the length of a missing side when two sides and the angle contained between them is given. It is also used to find missing angles when all three sides are known. Finding the Length of a Missing Side Finding the Size of a Missing Angle The Sine Rule The Sine Rule is used to find the length of a missing side when two angles and one side length are given. It is also used to find the size of a missing angle when two sides are given and the angle opposite one of the given sides is also given. Finding the Size of a Missing Angle Finding the Length of a Missing Side One must be careful in using the Sine Rule to find an unknown angle. If sin θ = 0.5 then θ = 30 or θ = 150. The following example will illustrate the caution. Example 4 In ABC, AB = 8, BC = 10 and ACB ˆ = 30. Determine the size of CAB. ˆ
Example 5 (Example 3 Revisited) Two observers Sona and Yoro, 20 m apart observe a kite in the same vertical plane and from the same side of the kite. The angles of elevation from Sona and Yoro are 25 and 55 respectively. Determine the height of the kite to the nearest metre. Bearings A bearing is used to represent the direction of one point relative to another point. bearing is measured in a clockwise direction from North. A Example 6 Y is 25 km from X on a bearing of 075. Determine the bearing to X from Y.
Example 7 On a bearing of 046, it is 8.1 km from Senegambia to Bakau. On a bearing of 122, it is 10.6 km from Senegambia to Abuko. a) Determine the direct distance from Bakau to Abuko. b) Determine the bearing from Bakau to Abuko.
Example 8 Recently on a morning walk, John headed out on a bearing of 045 and walked 5 km. At that point he changed his direction to a bearing of 110 and walked a further 3 km. a) Determine John s distance from his original starting point. b) Determine the bearing that John must take to walk directly to his original starting point.
Great and Small Circles A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the sphere into two equal hemispheres. A small circle of a sphere is the circle constructed by a plane crossing the sphere, parallel to a great circle, but not through the centre of the sphere. Small circles always have a smaller radius than the sphere. When an orange is sliced in half, two hemispheres are created and the flat bottom of one of the hemispheres is an example of a great circle. If you make slices parallel to the bottom of the hemisphere, the radius of each slice would get smaller than that of the previous slice. These slices would illustrate small circles. The Earth is generally considered to be spherical in shape with a radius of approximately 6 400 km. The Earth rotates about a polar axis, a straight line through the Earth s centre, which goes through the north and south poles. When great circles are drawn through the north and south poles, the lines formed on the surface of the Earth are called meridian lines or lines of longitude. The Greenwich Meridian is used as the standard from which other meridians are measured up to 180 W and 180 E. The Greenwich Meridian is also known as the Prime Meridian. When circles are drawn perpendicular to the polar axis, the circles formed on the surface of the Earth are called lines of latitude. The line of latitude whose centre is also the centre of the Earth is a great circle called the Equator. All other lines of latitude are small circles. These circles are called parallels of latitude and are measured in degrees north or south of the equator. Lines of latitude vary from 90 N to 90 S. The angle formed by the radius drawn from the centre of the Earth to the Equator and the centre of the Earth to the circumference of the small circle determines the degrees of the parallel of latitude. If it is north of the Equator, it is measured in N and if it is south of the Equator, it is measured in S. In the above diagram, OQ is the radius of the Earth drawn to the Equator, OA is the radius of the Earth drawn to the parallel of latitude at 42 S that passes through A.
Example 9 Determine the length of the parallel of latitude 42 S. Also determine how fast a person standing at point A is travelling as a result of the Earth s rotation. Assume the radius of the Earth is approximately 6 400 km and the circumference of the Earth is 40 000 km. A Key Idea Before Proceeding AB is an arc on the circumference of the circle with centre C, shown to the right. The angle at the centre of the circle is called a sector angle. θ The length of the arc is 2πr where r is the radius of the 360 circle and θ is the sector angle. Notice that the arc length could also be thought of as the length subtended by the reflex angle, 360 θ. We are concerned with the shortest arc length.
Example 10 Both Quito, Ecuador and Mbandaka, DR of Congo are located on the Equator. Quito is at 78.5 W longitude and Mbandaka is at 18.5 E longitude. Determine the distance from Quito to Mbandaka along the Equator. Use 6 400 km as the radius of the Earth. Example 11 Both Cape Town, South Africa and Mbandaka, DR of Congo lie on longitude 18.5 E. Their latitudes are 34 S and 0 N respectively. Calculate the shortest distance between Cape Town and Mbandaka. Use 40 000 km as the circumference of the Earth.
Example 12 Banjul, The Gambia and Mek ele, Ethiopia lie along latitude 13.3 N. Banjul is located at (13.3 N, 16.6 W) and Mek ele is located at (13.3 N, 39.6 E). If an airplane flies along latitude 13.3 N from Banjul to Mek ele averaging 525 km/h, how long would the flight take? Use 40 000 km as the circumference of the Earth.
Example 13 Barsakelmes Nature Reserve, near the Aral Sea, in Kazakhstan is located at 46.3 N latitude and 59.6 E longitude. Gilford Pinchot National Forest in Washington State, USA is located at 46.3 N latitude and 120.4 W longitude. a) Determine the distance between the nature reserve and the forest along the parallel of latitude. b) Determine the distance between the nature reserve and the forest along a great circle.