FBE05 – Mathematics and Statistics Lecture 5 - Triangles

Introduction The word trigonometry comes from the two Greek words trigonon (triangle) and metria (measure). Trigonometry may be defined as a branch of mathematics that deals with the study of relationships between the sides and the angles of triangles (Δ). The complex history of the term ‘sine’ reveals that the origins of trigonometry trace to the ancient cultures of Egyptian, Babylonian, Greek and Indus Valley civilisations. Originally, the use of trigonometry became popular as developments started to take place in astronomy. To calculate the position of the planets the astronomers used concepts we now refer to as trigonometry. The use of trigonometry is not just limited to mathematics. It is also used in physics, land surveying, engineering, satellite navigation and other applications. may be defined as the space in a three-dimensional object. This is different from area, which is applicable to two-dimensional shapes. The most basic shapes in volume calculations are the cube and the cuboid. A cube is a three-dimensional figure which has six square faces. This means that the length, width and height are equal in a cube. In a cuboid, at least one of the sides will be different from the others. Typical example of a cuboid is an aerated concrete block.

The trigonometrical ratios Consider a right-angled triangle CAB, shown in Figure 4.1. Angle CBA (<B) is a right angle, i.e. 90°. If <A is being considered, side BC is called the ‘opposite side’ or opposite. Side AC, which is the longest side, is called the ‘hypotenuse’. The third side, AB, is called the ‘adjacent side’ or adjacent and is common to <A as well as the right angle. The ratios of the sides of a right-angled triangle, called the ‘trigonometrical ratios’, are sine, cosine and tangent. In ΔCAB (Figure 4.1):

The trigonometrical ratios Figure 4.1

The trigonometrical ratios

The trigonometrical ratios The 30°, 60°, 90° and 45°, 45°, 90° triangles are shown in Figure 13.3. Figure 13.3a shows a 30°, 60°, 90° triangle whose sides are in the ratio 1:2: 3 (or 1:2:1.732). If side AC = 5 cm, then AB, being two times. AC, is equal to 10 cm. Using the theorem of Pythagoras, BC = 8.66 cm:

The trigonometrical ratios Figure 13.3

The trigonometrical ratios

The trigonometrical ratios Determine angles A and B of ΔABC shown in Figure 13.4 Figure 13.4

The trigonometrical ratios Find side AB of the triangle shown in Figure 13.5. Figure 13.5

The trigonometrical ratios A flat roof of 3 m span has a fall of 75 mm. Find the pitch of the roof.

The trigonometrical ratios The gradient of a road is 1 in 5. Find the angle that the road makes with the horizontal. A surveyor, 100 m from a building, measures the angle of elevation to the top of a building to be 40°. If the height of the instrument is 1.400 m and the ground between the surveyor and the building is level, find the height of the building.

The trigonometrical ratios Jane, standing on the 15th floor of a building, looks at her car parked on the nearby road. Find the angle of depression if the other details are as shown in Figure 13.10. Figure 13.10

The trigonometrical ratios-Stairs Stairs in buildings are provided for access from one floor to the other. The Building Regulations give guidelines on their design for use in domestic and commercial buildings. According to the Building Regulations 1995: (i) The pitch of stairs in dwelling houses should not be more than 42° (ii) Maximum rise (R) = 220mm (iii) Minimum going (G) = 220mm (iv) 2 x Rise + Going (or 2R + G) must be within 550 and 700mm (v) The headroom should not be less than 2.0m

The trigonometrical ratios-Stairs Example: A flight of stairs has 12 steps. Each step has a rise of 210 mm and going equal to 230 mm: (a) Find the pitch of the stairs (b) If the pitch exceeds the limit set by the Building Regulations, find the satisfactory dimensions of the rise and going

The trigonometrical ratios-Stairs Example: The floor to floor height in a house is 2.575 m and the space to be used for providing the stairs is shown in Figure 13.13. Design a staircase that satisfies the requirements of the Building Regulations.

The trigonometrical ratios Example: The roof shown in Figure 13.16 has a height of 4 m and a span of 10 m. Calculate: (a) Pitch of the roof (b) True lengths of common rafters (c) Surface area of the roof (d) The number of single lap tiles required to cover the roof

The trigonometrical ratios Figure 13.16

The trigonometrical ratios Example The pitch of a 14 m long hipped roof is 45°. If other dimensions are as shown in Figure 13.18, find: (a) Height of the roof (b) Length of common rafter XZ (c) True length of hip rafter DA

The trigonometrical ratios Figure 13.18