BY : MUSHTAQ-UR-REHMAN HEAD OF MATHEMATICS DEPARTMENT D.A.P.S.
O & A LEVELS Trigonometry and Applications
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Indeed we have created everything in a proper measure. (Surah
Al-Qamr) Allah says in the Holy Quran
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Trigonometry and Applications Fields of discussion What is
Mathematics? Prince of Mathematicians What is Trigonometry? History
and the meaning of the word sine and cosine. Trigonometric
functions, Circular functions or cyclometric functions Fields of
Trigonometry Ancient Egypt and the Mediterranean world Applications
of Trigonometry Angle measurement Properties of sines and cosines
The Law(Rule) of sines,cosines Trigonometric Equations Applications
of Trigonometric Equations
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What is Mathematics? Etymology The word Mathematics" comes from
the Greek word (mthma), which means learning, study, science, and
additionally more technical meaning Mathematical study",
Mathematics (Definition) A group of related subjects, including
ALGEBRA, GEOMETRY, TRIGONOMETRY and CALCULUS, concerned with the
study of number,quantity, structure, shape and space. Applications
Mathematics is used throughout the world as an essential tool in
many fields, including natural science, engineering, medicine, and
the social sciences.
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Prince of Mathematicians Carl Friedrich Gauss Himself known as
the "prince of mathematicians, referred to Mathematics as "the
Queen of the Sciences ". This German Mathematician contributed to
many areas of Mathematics, including probability theory, algebra,
and geometry. He proved that every polynomial has at least one
root, or solution; this theory is known as the fundamental theory
of algebra. Gauss also applied his mathematical work to theories of
electricity and magnetism.
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What is Trigonometry? Etymology The word Trigonometry is
derived from three Greek words tries(three), goni(angle) and
metron(measurement). So literally, this word means measurement of
the triangle. Trigonometry (Definition) The branch of Mathematics
concerned with the properties of trigonometric functions and their
application to the determination of the sides and angles of
triangles. Trigonometry has now a wide application in higher
Mathematics in fact, any attempt to study Higher Mathematics would
be an utter failure without a working knowledge of trigonometry. It
has applications in both pure mathematics and applied mathematics,
where it is essential in many branches of science and
technology.
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History and the meaning of the word sine and cosine Interesting
word history for "sine The Hindu mathematician Aryabhata (about
475550 A.D.) used the Sanskrit word jya or jiva for the half-chord
which was sometimes shortened to jiva. This was brought into Arabic
as jiba, and written in Arabic simply with two consonants jb,
vowels not being written. Later, Latin translators selected the
word sinus to translate jb thinking that the word was an arabic
word jaib, which meant bosom, fold, or bay, The Latin word for
bosom, bay, or curve is sinus. In English, sinus was imported as
"sine". This word history for "sine" is interesting because it
follows the path of trigonometry from India, through the Arabic
language from Baghdad through Spain, into western Europe in the
Latin language, and then to modern languages such as English.
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Trigonometric functions, Circular functions or cyclometric
functions Any of a group of functions expressible in terms of the
ratios of the sides of right-angled triangle. Sine Ratio The sine
of an angle in a right triangle equals the opposite side divided by
the hypotenuse: sin =opp/hyp Cosine Ratio. Cosines are just sines
of the complementary angle. Thus, the name "cosine" ("co" being the
first two letters of "complement"). The complementary angle equals
the given angle subtracted from a right angle, 90. For instance, if
the angle is 30, then its complement is 60. Generally, for any
angle x, cos =adj/hyp cos x = sin (90 x). Or cos 50 = sin (90 50) =
sin40 Tangent Ratio tanx = sinx/cosx tanx = opp/adj
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Trigonometric functions, Circular functions or cyclometric
functions Secant : sec q = 1/cos q Cosecant: csc q = 1/sin q
Cotangent: cot q = 1/ tan q cot q = cos q/sin q tan q = sin q/cos
q
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Trigonometric Ratios
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Trigonometric Identities The following formulas, called
identities, which show the relationships between the trigonometric
functions, hold for all values of the angle , or of two angles, and
, for which the functions involved are
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Fields of Trigonometry Plane Trigonometry In many applications
of trigonometry the essential problem is the solution of triangles.
If enough sides and angles are known, the remaining sides and
angles as well as the area can be calculated, and the triangle is
then said to be solved. Triangles can be solved by the law of sines
and the law of cosines. Surveyors apply the principles of geometry
and trigonometry in determining the shapes, measurements and
position of features on or beneath the surface of the Earth. Such
topographic surveys are useful in the design of roads, tunnels,
dams, and other structures.
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Fields of Trigonometry Spherical Trigonometry Spherical
Trigonometry involves the study of spherical triangles, which are
formed by the intersection of three great circle arcs on the
surface of a sphere. Great Circle A great circle is a theoretical
circle, such as the equator, formed by the intersection of the
earths surface and an imaginary plane that passes through the
center of the earth and divides it into two equal parts. Navigators
use great circles to find the shortest distance between any Two
points on the earths surface.
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Fields of Trigonometry Analytic Trigonometry Analytic
Trigonometry combines the use of a coordinate system, such as the
Cartesian coordinate system used in analytic geometry, with
algebraic manipulation of the various trigonometry functions to
obtain formulas useful for scientific and engineering
applications.
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Ancient Egypt and the Mediterranean world Several ancient
civilizationsin particular, the Egyptian, Babylonian, Hindu, and
Chinesepossessed a considerable knowledge of practical geometry,
including some concepts of trigonometry. A close analysis of the
text, with its accompanying figures, reveals that this word means
the slope of an incline, essential knowledge for huge construction
projects such as the pyramids. It shows that the Egyptians had at
least some knowledge of the numerical relations in a triangle, a
kind of proto- trigonometry.
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Ancient Egypt and the Mediterranean world
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Applications of Trigonometry. Fields that use trigonometry or
trigonometric functions include Astronomy (especially for locating
apparent positions of celestial objects(star or planet), in which
spherical trigonometry is essential) and hence navigation (on the
oceans, in aircraft, and in space), to measure distances between
landmarks, and in satellite navigation systems. The sine and cosine
functions are fundamental to the theory of periodic functions such
as those that describe sound and light waves. Music theory,
acoustics(study of sound ), optics, electronics, probability
theory, statistics, biology, medical imaging (CAT scans and
ultrasound), pharmacy, chemistry, number theory cryptology(coding),
seismology, meteorology, oceanography, many physical sciences, land
surveying and geodesy(cartography), architecture, phonetics (sounds
of human speech), economics, electrical engineering, mechanical
engineering, civil engineering, computer graphics, crystallography
and game development.
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Applications of Trigonometry Marine sextants like this are used
to measure the angle of the sun or stars with respect to the
horizon. Using trigonometry and a marine chronometer(timer), the
position of the ship can then be determined from several such
measurements.
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Applications of Trigonometry Wave Mathematics Waves are
familiar to us from the ocean, the study of sound, earthquakes, and
other natural phenomenon. Ocean waves come in very different sizes
to fully understand waves, we need to understand measurements
associated with these waves, such as how often they repeat (their
frequency), and how long they are (their wavelength), and their
vertical size (amplitude). The importance of the sine and cosine
functions is in describing periodic phenomenathe vibrations of a
violin string, the oscillations of a clock pendulum, or the
propagation of electromagnetic waves, sound and light waves.
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Applications of Trigonometry Sine waves in nature i)Sound waves
are sine waves whenever we listen to music, we are actually
listening to sound waves. ii) light waves are also sine waves.
iii)Radio waves are sine waves. iv)Simple harmonic motion of a
spring when pulled and released is a sine wave. v) Alternating
current (AC) is a sine wave. vi) Pendulum clock oscillations are
sinusoidal in nature vii) Waves of ocean are sinusoidal. viii) The
vibrations of guitar strings when played are sinusoidal in
nature.
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Applications of Trigonometry Graph of Trigonometric Functions
Graph of sine function f(x) = a sin ( bx + c ) Graph of sine
function f(x) = a cos ( bx + c ) Graph of tangent function f(x) =
tanx
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Applications of Trigonometry The 17th and 18th centuries saw
the invention of numerous mechanical devices. A notable application
was the science of artilleryand in the 18th century it was a
science. Galileo Galilei (15641642) discovered that any motionsuch
as that of a projectile under the force of gravitycan be resolved
into two components, one horizontal and the other vertical, This
discovery led scientists to the formula for the range of a
cannonball when its muzzle velocity v 0 (the speed at which it
leaves the cannon) and the angle of elevation A of the cannon.
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Applications of Trigonometry
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Fourier series An infinite trigonometric series of terms
consisting of constants multiplied by sines or cosines, used in the
approximation of periodic functions. The trigonometric or Fourier
series have found numerous applications in almost every branch of
science, from optics and acoustics to radio transmission and
earthquake analysis. Their extension to non periodic functions
played a key role in the development of quantum mechanics in the
early years of the 20th century. Trigonometry, by and large,
matured with Fourier's theorem.
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Angle measurement The concept of angle is one of the most
important concepts in geometry and the subject of trigonometry is
based on the measurement of angles. Degree (Angle) There are two
commonly used units of measurement for angles. The more familiar
unit of measurement is that of degrees. A circle is divided into
360 equal degrees, Degrees may be further divided into minutes and
seconds. For instance seven and a half degrees is now usually
written 7.5. Each degree is divided into 60 equal parts called
minutes. So seven and a half degrees can be called 7 degrees and 30
minutes, written 7 30'. Each minute is further divided into 60
equal parts called seconds, and, for instance, 2 degrees 5 minutes
30 seconds is written 2 5' 30".
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Angle measurement Radian(Angle) The other common measurement
for angles is radians. If the radius of the circle and the length
of arc of a sector of the circle are equal then angle is 1 radian.
The radian measure of the angle is the ratio of the length of the
subtended arc to the radius of the circle. radian measure = arc
length/radius ( = S/r) Below is a table of common angles in both
degree measurement and radian measurement. DegreesRadians 90 /2 60
/3 45 /4 30 /6
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Angle measurement
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1. Express the following angles in radians. (a). 12 degrees, 28
minutes, that is, 12 28'. (b). 36 12'. 2. Reduce the following
numbers of radians to degrees, minutes, and seconds. (a). 0.47623.
(b). 0.25412. 3. Given the angle a and the radius r, to find the
length of the subtending arc. a = 0 17' 48", r = 6.2935. 4. Find
the length to the nearest inch of a circular arc of 11 degrees 48.3
minutes if the radius is 3200 feet. 5. Given the length of the arc
l and the angle a which it subtends at the center, to find the
radius. a = 0 44' 30", l =.032592
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Properties of sines and cosines 1.Sine and cosine are periodic
functions of period 360 or 2 ., sin (t + 360) = sin t, and sin (t +
2 ) = sin t, cos (t + 360) = cos t. cos (t + 2 ) = cos t. 2. Sine
and cosine are complementary: cos t = sin ( /2 t), sin t = cos ( /2
t) 3.The Pythagorean identity sin 2 t + cos 2 t = 1. 4. Sine is an
odd function, and cosine is even sin (t) = sin t, and cos (t) = cos
t. 5.An obvious property of sines and cosines is that their values
lie between 1 and 1. Every point on the unit circle is 1 unit from
the origin, so the coordinates of any point are within 1 of 0 as
well.
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The Law(Rule) of sines The Law of Sines is simple and beautiful
and easy to derive. Its useful when you know two angles and any
side of a triangle, or sometimes when you know two sides and one
angle. Law of Sines First Form: a / sin A = b / sin B = c / sin C
This is very simple and beautiful: for any triangle, if you divide
any of the three sides by the sine of the opposite angle, youll get
the same result. This law is valid for any triangle. Law of Sines
Second Form: sin A / a = sin B / b = sin C / c
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The Law(Rules) of cosines The Law of Sines is fine when you can
relate sides and angles. But suppose you know three sides of the
triangle for instance a = 180, b = 238, c = 340 and you have to
find the three angles. The Law of Sines is no good for that because
it relates two sides and their opposite angles. If you dont know
any angles, you have an equation with two unknowns and you cant
solve it. Law of Cosines First Form: cos A = (b + c a) / 2bc cos B
= (a + c b) / 2ac cos C = (a + b c) / 2ab Law of Cosines Second
Form: a = b + c 2bc cos A b = a + c 2ac cos B c = a + b 2ab cos
C
Slide 36
Trigonometric Equations A formula that asserts that two
expressions have the same value ;it is either an identical equation
or an identity which is true for any values of the variables or a
conditional equation which is only true for certain values of the
variables. Example1 Solve the equation sin{1/3( -30) } = 3/2,
giving all the roots in the interval 0 360 . Example2 Find all the
values of in the interval 0 360 for which sin2 = cos36 Example3
Find all the values of in the interval 0 360 for which sin2 - 3
cos2 = 0
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Applications of Trigonometric Equations Example The height in
meters of the water in a harbor is given by approximately by the
formula d=6+3cos30t where t is the time measured in hours from
noon. Find the time after noon when the height of the water is 7.5
meters for the second time.
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THANK YOU Mushtaq ur Rehman H.O.D Mathematics Department
D.A.P.S. O&A Levels