Trigonometry Basics

by Satish

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 $y=\frac{x-1}{x+1},$ 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,
$\frac{a}{h},\frac{b}{h},\frac{a}{b},$
and their reciprocals
$\frac{h}{a},\frac{h}{b},\frac{b}{a},$
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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.

$\frac{a}{h} \quad i.e \quad \frac{side \quad opposite\quad angle \quad A}{hypotenuse}$

is called Sine of angle A

$\frac{b}{h} \quad i.e \quad \frac{side \quad adjacent\quad angle \quad A}{hypotenuse}$

is called Cosine of angle A

$\frac{a}{b} \quad i.e \quad \frac{side \quad opposite\quad angle \quad A}{side \quad adjacent\quad angle \quad A}$

is called Tangent of angle A

$\frac{h}{a}$

i.e the reciprocal of the Sine is called Cosecant of angle A

$\frac{h}{b}$

i.e the reciprocal of the Cosine is called Secant of angle A

$\frac{b}{a}$

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

$SinA = \frac{BC}{AB}=\frac{4}{5}$
$CosA = \frac{AC}{AB}=\frac{3}{5}$
$TanA = \frac{BC}{AC}=\frac{4}{3}$
$CosecA = \frac{1}{SinA}=\frac{5}{4}$
$SecA = \frac{1}{CosA}=\frac{5}{3}$
$CotA = \frac{1}{TanA}=\frac{3}{4}$

Basic Identities

$Sinx=\frac{1}{Cosecx}$	$Cosx = \frac{1}{Secx}$	$Tanx=\frac{1}{Cotx}$
$Cosecx = \frac{1}{Sinx}$	$Secx = \frac{1}{Cosx}$	$Cotx= \frac{1}{Tanx}$
$Tanx=\frac{Sinx}{Cosx}$	$Cotx =\frac{Cosx}{Sinx}$

Phythagorean Identities	Symmetry Properties
	$Cos(-x) = Cosx \quad Sec(-x) = Secx$ $Sin(-x) = -Sinx \quad Cosce(-x) = -Cosecx$ $Tan(-x) = -Tanx \quad Cot(-x) = -Cotx$