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Trigonometry
Introduction
Trigonometry is a branch of mathematics that deals with the study of angles and their relationships with
sides of triangles
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Topics in Trigonometry:

Basic concepts of trigonometry

Trigonometric functions

Trigonometric identities

Inverse trigonometric functions

Applications of trigonometry

Solving trigonometric equations

Basic Concepts of Trigonometry:

a
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The measure of an angle is usually
expressed in degrees or radians
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b
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The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs
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Trigonometric ratios: Trigonometric ratios are ratios of the lengths of the sides of a right triangle
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They are defined as follows:

Sine: sin(theta) = opposite/hypotenuse
Cosine: cos(theta) = adjacent/hypotenuse
Tangent: tan(theta) = opposite/adjacent
d
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In other words,
a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the legs
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Sine function: The sine function is defined as the ratio of the length of the opposite side to the length
of the hypotenuse of a right triangle
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The sine function is periodic with a
period of 2π, which means that it repeats itself every 2π units
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Cosine function: The cosine function is defined as the ratio of the length of the adjacent side to the
length of the hypotenuse of a right triangle
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The cosine function is also
periodic with a period of 2π
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Tangent function: The tangent function is defined as the ratio of the length of the opposite side to the
length of the adjacent side of a right triangle
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The tangent function is not
periodic
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Cosecant function: The cosecant function is the reciprocal of the sine function
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The cosecant function is not periodic
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Secant function: The secant function is the reciprocal of the cosine function
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The secant function is not periodic
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Cotangent function: The cotangent function is the reciprocal of the tangent function
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The cotangent function is periodic with a period of π
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Pythagorean identities: The Pythagorean identities are a set of identities that relate the sine, cosine,
and tangent functions to each other
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The three
Pythagorean identities are:

sin^2(theta) + cos^2

tan^2(theta) + 1 = sec^2(theta)
1 + cot^2(theta) = csc^2(theta)

b
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They are as follows:

csc(theta) = 1/sin(theta)
sec(theta) = 1/cos(theta)
cot(theta) = 1/tan(theta)
sin(theta) = 1/csc(theta)
cos(theta) = 1/sec(theta)
tan(theta) = 1/cot(theta)

c
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They are as follows:

tan(theta) = sin(theta)/cos(theta)
cot(theta) = cos(theta)/sin(theta)
sec(theta) = 1/cos(theta)
csc(theta) = 1/sin(theta)
d
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They are as follows:
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sin(-theta) = -sin(theta) (odd)
cos(-theta) = cos(theta) (even)
tan(-theta) = -tan(theta) (odd)
csc(-theta) = -csc(theta) (odd)
sec(-theta) = sec(theta) (even)
cot(-theta) = -cot(theta) (odd)
Inverse Trigonometric Functions:
a
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The domain of the inverse sine function is [-1, 1], and the range is [-π/2, π/2]
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Inverse cosine function: The inverse cosine function is denoted by cos^-1(x) and is defined as the
angle whose cosine is x
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c
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The domain of the inverse tangent function is (-∞, ∞), and the range is (-π/2,
π/2)
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Inverse cotangent function: The inverse cotangent function is denoted by cot^-1(x) and is defined as
the angle whose cotangent is x
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Applications of Trigonometry:
a
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It is used in various fields such as aviation, marine navigation, and land surveying
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Astronomy: Trigonometry is used in astronomy to calculate the position of celestial bodies and to
study the motion of planets and stars
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Engineering: Trigonometry is used in engineering to design and build structures such as bridges and
buildings
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Physics: Trigonometry is used in physics to study waves, vibrations, and oscillations
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Linear equations: Linear trigonometric equations can be solved using algebraic methods such as
substitution and simplification
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Quadratic equations: Quadratic trigonometric equations can be solved by factoring, completing the
square, or using the quadratic formula
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Other types of equations

: Other types of trigonometric equations, such as equations involving multiple angles or equations
involving inverse trigonometric functions, can be solved using various techniques such as the sum and
difference formulas, the double angle formula, or the use of trigonometric identities
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The unit circle: The unit circle is a circle with a radius of one unit centered at the origin of a coordinate
plane
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b
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The graphs of these functions exhibit
periodic behavior and have certain properties such as amplitude, period, and phase shift
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Transformations of trigonometric functions: Trigonometric functions can be transformed by shifting,
stretching, or reflecting their graphs
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Trigonometric Identities and Equations:
a
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b
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Applications of Trigonometry in Real Life:
Trigonometry has numerous applications in real life, some of which are listed below:

a
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b
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c
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d
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e
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f
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Conclusion :
Trigonometry is an important branch of mathematics that deals with the study of triangles and their
properties
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The study of trigonometry involves learning various concepts such as the
trigonometric functions, identities, equations, and their applications
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