Trigonometric Functions

Trigonometry
Trigonometry is the branch of mathematics that deals with the relationship between sides and angles of a right-angled triangle. the name of these relations are sine, cosine, tangent, cotangent, secant, cosecant. 
There are ratios used to study this relationship are called trigonometric ratios, namely, sine, cosine, tangent, cotangent, secant, cosecant. The word trigonometry is a 16th century Latin derivative and the concept was given by the Greek mathematician Hipparchus. 

Introduction to Trigonometry
The word trigonometry is formed by combining two words 'Trigonon' and 'Metron' which means triangle and measure respectively. It is the study of the relation between the sides and angles of a right-angled triangle.

In a right-angled triangle, we have the following three sides.

Perpendicular - It is the side opposite to the angle θ.
Base - This is the adjacent side to the angle θ.
Hypotenuse - This is the side opposite to the right angle.

trigonometric Ratios

There are basic six ratios in trigonometry that help in establishing a relationship between the ratio of sides of a right triangle with the angle. If θ is the angle in a right-angled triangle, formed between the base and hypotenuse, then

• sin θ = Opposite Side/Hypotenuse
• cos θ = Adjacent Side/Hypotenuse
• tan θ = Opposite Side/Adjacent Side
• cot θ = 1/tan θ = Adjacent Side/Opposite Side
• sec θ = 1/cos θ = Hypotenuse/Adjacent Side
• cosec θ = 1/sin θ = Hypotenuse/Opposite Side

The value of the other three functions: cot, sec, and cosec depend on tan, cos, and sin respectively as given below.

• cot θ = 1/tan θ = Base/Perpendicualr
• sec θ = 1/cos θ = Hypotenuse/Base
• cosec θ = 1/sin θ = Hypotenuse/Perpendicular

What are Trigonometric Functions?

There are six basic trigonometric ratios used in Trigonometry. These ratios are trigonometric functions. The six basic trigonometric functions are sine function, cosine function, secant function, co-secant function, tangent function, and co-tangent function. 

Unit Circle and Trigonometric Values

Unit circle can be used to calculate the values of basic trigonometric functions- sine, cosine, and tangent. The following diagram shows how trigonometric ratios sine and cosine can be represented in a unit circle.

Trig Functions in Four Quadrants

Trigonometric Functions Graph

The graphs of trigonometric functions have the domain value of θ represented on the horizontal x-axis and the range value represented along the vertical y-axis.

Domain and Range of Trigonometric Functions

The value of θ represents the domain of the trigonometric functions and the resultant value is the range of the trigonometric function. The domain values of θ are in degrees or radians and the range is a real number value. Generally, the domain of the trigonometric function is a real number value, but in certain cases, a few angle values are excluded because it results in a range as an infinite value. The trigonometric function are periodic functions. The below table presents the domain and range of the six trigonometric functions.

Trigonometric Functions	Domain	Range
Sinθ	(-∞, + ∞)	[-1, +1]
Cosθ	(-∞ +∞)	[-1, +1]
Tanθ	R - (2n + 1)π/2	(-∞, +∞)
Cotθ	R - nπ	(-∞, +∞)
Secθ	R - (2n + 1)π/2	(-∞, -1] U [+1, +∞)
Cosecθ	R - nπ	(-∞, -1] U [+1, +∞)

Trigonometric Functions Identities

The trigonometric functions identities are broadly divided into reciprocal identities, Pythagorean formulas, sum and difference of trig functions identities, formulas for multiple and sub-multiple angles, sum and product of identities. All of these below formulas can be easily derived using the ratio of sides of a right-angled triangle. The higher formulas can be derived by using the basic trigonometric function formulas. Reciprocal identities are used frequently to simplify trigonometric problems.

Reciprocal Identities

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

Pythagorean Identities

• Sin²θ + Cos²θ = 1
• 1 + Tan²θ = Sec²θ
• 1 + Cot²θ = Cosec²θ

Sum and Difference Identities

• sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
• cos(x+y) = cos(x)cos(y) – sin(x)sin(y)
• tan(x+y) = (tan x + tan y)/ (1−tan x tan y)
• sin(x–y) = sin(x)cos(y) – cos(x)sin(y)
• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)
• tan(x−y) = (tan x–tan y)/ (1+tan x tan y)

Half-Angle Identities

• sin A/2 = ±√[(1 - cos A) / 2]
• cos A/2 = ±√[(1 + cos A) / 2]
• tan A/2 = ±√[(1 - cos A) / (1 + cos A)] (or) sin A / (1 + cos A) (or) (1 - cos A) / sin A

Double Angle Identities

• sin(2x) = 2sin(x) cos(x) = [2tan x/(1+tan² x)]
• cos(2x) = cos²(x)–sin²(x) = [(1-tan² x)/(1+tan² x)]
• cos(2x) = 2cos²(x)−1 = 1–2sin²(x)
• tan(2x) = [2tan(x)]/ [1−tan²(x)]
• cot(2x) = [cot²(x) - 1]/[2cot(x)]
• sec (2x) = sec² x/(2-sec² x)
• cosec (2x) = (sec x. cosec x)/2

Triple Angle Identities

• Sin 3x = 3sin x – 4sin³x
• Cos 3x = 4cos³x - 3cos x
• Tan 3x = [3tanx-tan³x]/[1-3tan²x]

Product identities

• 2sinx⋅cosy=sin(x+y)+sin(x−y)
• 2cosx⋅cosy=cos(x+y)+cos(x−y)
• 2sinx⋅siny=cos(x−y)−cos(x+y)

Sum of Identities

• sinx+siny=2sin((x+y)/2) . cos((x−y)/2)
• sinx−siny=2cos((x+y)/2) . sin((x−y)/2)
• cosx+cosy=2cos((x+y)/2) . cos((x−y)/2)
• cosx−cosy=−2sin((x+y)/2 . sin((x−y)/2)

Class 8 Maths NCERT Solutions Chapter 1 to 16

Chapter 3 Trigonometric Functions (Video Lectures)
Lecture 1 Introduction to trigonometric functions
Lecture 2 Trigonometric Function as circular Functions
Lecture 3  Understanding Quadrilaterals
Chapter 4 Practical Geometry
Chapter 5 Data Handling
Chapter 6 Squares And Square Roots
Chapter 7 Cubes And Cube Roots
Chapter 8 Comparing Quantities
Chapter 9 Algebraic Expressions And Identities
Chapter 10 Visualising Solid Shapes
Chapter 11 Mensuration
Chapter 12 Exponents And Powers
Chapter 13 Direct And Inverse Proportions
Chapter 14 Factorisation
Chapter 15 Introduction To Graphs
Chapter 16 Playing With Numbers

US  standard measurement	Metrics measurement
1 in	2.54 cm
1 ft	0.3048 m
1 yd	0.914 m
1 mile	1.609 km

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