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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 time-saving 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.
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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.
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.
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Solution: The hypotenuse of the triangle
equals 5. Hence, applying the definitions, we have |
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Basic
Identities
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| Phythagorean Identities | Symmetry Properties |
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Graphs of the Six Trigonometric
Functions
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