Definition of
Trigonometry: Trigonometry considers the properties
of angles and certain ratios associated with angles, and applies
the knowledge of these properties to the solution of triangles and
various other algebraic and geometric problems. Incidentally
trigonometry considers also certain timesaving aids in computation
such as logarithms, which are generally employed in the solution of
triangles. Briefly stated,
Trigonometry is the science of angular magnitudes and the art
of applying the principles of this science to the solution of
problems.
The word Trigonometry comes from two
Greek words, trigonon = triangle, and metron = measure. The method
was originated in the second century B.C. by Hipparchus and other
early Greek astronomers in their attempts to solve certain
spherical triangles. The term trigonometry was not used until the
close of the sixteenth century.
Before we get into the basic
definitions of Trigonometric Functions, let us look at the basic
definition of a function_.
Definition of Function: When two
variables are so related that the value of the one depends upon the
value of the other, the one is said to be a function of the
other.
EXAMPLES: The area of a
square is a function of its side.
The volume of a sphere is a function of its radius.
The velocity of a falling body is a function of the time elapsed
since it began to fall.
The output of a factory is a function of the number of men
employed.
In the expression y depends upon x for its value, hence y is a
function of x.
Definition of Reciprocal: If the
product of two quantities equals unity, each is said to be the
reciprocal of the other.
For example, if xy = 1, x is the reciprocal of y, and y is the
reciprocalof x.
1/2 is the reciprocal of 2, and 2 is the reciprocal of 1/2, for
1/2X2=1.
In general, a/b and b/a are reciprocals since a/bxb/a=1 .
From xy = 1 it follows that x = 1/y, and y = 1/x, that is,
_The reciprocal of any quantity is unity divided by that
quantity.
Six Trigonometric Functions of an Acute
Angle: Let A be any acute angle, B any point on
either side of the angle, and ABC the
right triangle formed by drawing a perpendicular from B to the
other side of the angle. Denote AC, the side adjacent to the angle
A, by b (for base), BC, the side opposite the angle A, by a (for
altitude), and the hypotenuse AB by h.
The three sides of the right triangle form six different ratios,
namely,
and their reciprocals
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Since these ratios depend upon the
angle for their values, they are the functions of the angle
according to the general definition of a function that we discussed
at the beginning of our lesson. Each of these functions has
received a special name.
is called Sine of angle A
is called Cosine of angle A
is called Tangent of angle A
i.e the reciprocal of the Sine is called Cosecant of angle
A
i.e the reciprocal of the Cosine is called Secant of angle
A
i.e the reciprocal of the Tangent is called Cotangent of angle
A
The six functions just defined are
variously known as the trigonometric, circular, or
goniometric functions: trigonometric, because they
form the basis of the science of trigonometry; circular, because of
their relations to the arc of a circle; goniometric, because of
their use in determining angles, from gonia, a Greek word meaning
angle.
The terms sine of angle A, cosine of angle A, etc., are abbreviated
to sin A, cos A, tan A, cosec A , sec A , and cot A. The
definitions of the first six trigonometric functions must be
thoroughly memorized. The first three are especially important and
should be memorized.The remaining three functions may be remembered
most readily by the aid of the reciprocal relations, reciprocal
relations,
Sin A.Cosec A = 1
Cos A.Sec A = 1
Tan A.Cot A=1
It should be noticed that while a, b, and h are lines, the ratio of
any two of them is an abstract number; that is, the trigonometric
functions are abstract numbers. Also, the expressions sin A cos
A, tan A etc., are single symbols which cannot be separated,
sin has no meaning except as it is associated with some
angle.
EXAMPLE: The sides of a
right triangle are 3, 4, 5. Find all the trigonometric functions of
the angle A opposite the side 4.
Solution: The hypotenuse of the
triangle equals 5. Hence, applying the definitions, we have


Basic
Identities
Phythagorean
Identities 
Symmetry Properties 


Graphs of the Six Trigonometric
Functions
Image Credit: robindegrassi
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