UNIT1:Trigonometric Formulae and Equations
Introductory activity
The height h of the cathedral is 485 m. The angle of elevation
of the top of the Cathedral from a point 280 m away from the
base of its steeple on level ground is θ . By using trigonometricconcepts, find the value of this angle θ in degree.
Trigonometry studies relationship involving lengths and angles
of a triangle. The techniques in trigonometry are used for
finding relevance in navigation particularly satellite systems
and astronomy, naval and aviation industries, land surveying, in
cartography and (creation of maps) and medicine. Even if those
are the scientific applications of the concepts in trigonometry,
most of the mathematics we study would seem to have little reallife application.
Trigonometry is really relevant in our day to dayactivities.
Objectives
By the end of this unit, a student will be able to:
ᇢ Solve trigonometric equations.
ᇢ Use trigonometric formulae and equations inreal life.
1.1. Trigonometric formulae
Activity 1.1
Deduce the formula of
tan(A B + − ),sin(A B),cos(A B − − ), tan(A B)
Deduce the formula oftan(A B + − ),sin(A B),cos(A B − − ), tan(A B)
1.1.1. Addition and subtraction formulae
From activity 1.1:
The addition and subtraction formulae are
Addition and subtraction formulae are useful for finding
trigonometric number of some angles.
1.1.2. Double angle formulae
Activity 1.2
1.1.3. Half angle formulae
Activity 1.3
Solution
Solution
Example 1.10
Transform in product the sum cos 2 cos 4 x x
Activity 1.6
Activity 1.9
Discuss how trigonometric theory is used in refraction of light.
In optics, light changes speed as it moves from one medium to
another (for example, from air into the glass of the prism). This
speed change causes the light to be refracted and to enter the new
medium at a different angle (Huygens principle).
A prism is a transparent optical element with flat, polished surfacesthat refract light.
1.3.3. 1.3.3 Application in medicine
Activity 1.10
Conduct research in different books of the library or on the
internet to discover the application of trigonometry in the field
of medicine such as heart’s electrical activities, trucking the
rhythms of lungs capacity, testing electrical activities of the
brain and abnormalities in the brain...
Trigonometric concepts such as sinusoidal waves contribute to
various medical testing and interpretation of those test results.
1. Electrocardiography: The measurement of electrical activities
in the heart. Through this process, it is possible to determine
how long the electrical wave takes to travel from one part of
the heart to the next by showing if the electrical activity is
normal or slow, fast or irregular.
2. Pulmonary function testing: a spirometer is used to measure
the volume of air inhaled and exhaled while breathing by
recording the changing volume over time. The output of a
spirogram can be quantified using trigonometric equations
and generally, it is possible to describe any repeating rhythmsby a sine wave:
In humans, when each breath is completed, the lung still
contains a volume of air, called the functional residual capacity
(approximately 2200 mL in humans). Each inhalation adds from
500 mL of additional air for normal (resting) breathing. Each
exhalation removes approximately the same volume as was
inhaled. The volume of air inhaled and exhaled normally is called
the tidal volume. It takes approximately 5 seconds to complete a
single cycle. With this information, we can develop an equation for
lung capacity as a function of time for normal (resting) breathing.
Since sy describes the midpoint of the sine wave, we take the
functional residual capacity plus 1/2 the amplitude to give us
sy=2200+250=2450 mL. Our period is the time for a single cycle,
or p=5 seconds. The volume of air inhaled gives the amplitude, or
A=1000 mL. We therefore have the following equation for normalbreathing:
Unit Summary
1. The addition and subtraction formulae:
