Applied Math
Lesson 1
Lecturer: Dr. Mahdi Jalali
Trigonometry
Trigonometry is the branch of mathematics which deals with the
measurement of sides and angles of triangles, and their relationship
with each other. There are many applications in engineering where a
knowledge of trigonometry is needed.
The theorem of Pythagoras
OIn any right-angled triangle, the square on the hypotenuse is equal to
the sum of the squares on the other two sides.’
Problem 1. In Fig. 1, find the length of EF
By Pythagoras’ theorem:
Problem 2. Two aircraft leave an airfield at the same time. One travels due north at an
average peed of 300 km/h and the other due west at an average speed of 220 km/h.
Calculate their distance apart after 4 hours? Solution: After 4 hours, the first aircraft has
travelled 4x300=1200 km, due north, and the second aircraft has travelled 4x220=880
km due west
Trigonometric ratios
With reference to the right-angled triangle shown in the next Fig.
Problem 3.
Sine and cosine rules
To ‘solve a triangle’ means ‘to find the values of unknown sides and angles’. If a triangle is right angled,
trigonometric ratios and the theorem of Pythagoras may be used for its solution, as shown in previous section.
However, for a non-right-angled triangle, trigonometric ratios and Pythagoras’ theorem cannot be used.
Instead, two rules, called the sine rule and the cosine rule, are used.
Sine rule
With reference to triangle ABC of below Fig the sine rule states:
Cosine rule
With reference to triangle ABC of next Fig. the cosine rule states:
Area of any triangle:
Problem 5. In the next Fig. PR represents the inclined jib of a crane and is 10.0m long. PQ is
4.0m long. Determine the inclination of the jib to the vertical and the length of tie QR?
Coordinate Systems:
Cartesian Coordinate Systems: This type of coordinate system is the most commonly
used coordinate system in different calculations. These cartesian coordinate systems can
be used in both 2D and 3D positioning.
Polar Coordinate Systems: Polar coordinate systems are suitable for the 2D
positionings. It has a different logic other than cartesian coordinate systems. The values
to define the position of a point are angle and radius. Relative and absolute positionings
are possible for polar coordinate systems also.
Cylindrical Coordinate Systems: Cylindrical coordinate system is the same as the polar
coordinate system. But the cylindrical coordinate system is produced for 3D positioning
and it includes the radius and two angle values.
Spherical Coordinate Systems: Spherical coordinate system is another type of
coordinate system developed for 3D space positioning. It includes radius and two angle
values like cylindrical coordinate systems. But it has a different notation from
that. Coordinate systems are very common in CAD environments also.
Unite Circle
, tangent
areas and volumes of regular solids
The surface area of various solid shapes are given below:
Volume is the capacity of any solid shape. The formulae for volumes of various shapes are:
irregular areas and volumes:
Areas of irregular figures
Areas of irregular plane surfaces may be approximately determined by using
(a) a planimeter, (b) the trapezoidal rule, (c) the mid-ordinate rule, and (d)
Simpson’s rule. Such methods may be used, for example, by engineers
estimating areas of indicator diagrams of steam engines, surveyors
estimating areas of plots of land or naval architects estimating areas of
water planes or transverse sections of ships. (a)
A planimeter is an instrument for directly measuring small areas bounded
by an irregular curve.
(b) Trapezoidal rule: To determine the areas PQRS in the below Fig.
(c) Mid-ordinate rule: To determine the area ABCD of next Fig
(d) Simpson’s rule: To determine the area PQRS.
Solution:
(a) Trapezoidal rule:
(b) Mid-ordinate rule
(c) Simpson’s rule
Volumes of irregular solids: If the cross-sectional areas A1, A2, A3, . . . of
an irregular solid bounded by two parallel planes are known at equal
intervals of width d (as shown in next Fig), then by Simpson’s rule:
Problem. A tree trunk is 12m in length and has a varying cross-section. The
cross-sectional areas at intervals of 2m measured from one end are:
0.52, 0.55, 0.59, 0.63, 0.72, 0.84, 0.97 m2
