Trigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.
There are six basic trigonometric functions used in Trigonometry, which are:
- Sine Function (sin x)
- Cosine Function (cos x)
- Secant Function (Sec x)
- Cosecant Function (cosec x)
- Tangent Function (tan x)
- Cotangent Function (cot x)
The image added below shows a right-angle triangle PQR.
Then the six basic trigonometric functions formulas for this right-angle triangle are,
Function | Sides | Description | Relation |
|---|---|---|---|
sin θ | PQ/PR | Perpendicular/Hypotenuse | sin θ = 1/cosec θ |
cos θ | QR/PR | Base/Hypotenuse | cos θ = 1/sec θ |
tan θ | PQ/QR | Perpendicular/Base | tan θ = 1/cot θ |
sec θ | PR/QR | Hypotenuse/Base | sec θ = 1/cos θ |
cosec θ | PR/PQ | Hypotenuse/Perpendicular | cosec θ = 1/sin θ |
cot θ | QR/PQ | Base/Perpendicular | cot θ = 1/tan θ |
Identities
Below are the identities related to trig functions.
Even and Odd functions
The cos and sec functions are even functions; the rest other functions are odd functions.
- sin(-x) = -sin x
- cos(-x) = cos x
- tan(-x) = – tan x
- cot(-x) = -cot x
- csc(-x) = -csc x
- sec(-x) = sec x
Periodic Functions
The trig functions are the periodic functions. The smallest periodic cycle is 2π but for tangent and the cotangent it is π.
- sin(x+2nπ) = sin x
- cos(x+2nπ) = cos x
- tan(x+nπ) = tan x
- cot(x+nπ) = cot x
- csc(x+2nπ) = csc x
- sec(x+2nπ) = sec x
- Where n is any integer.
Pythagorean Identities
When the Pythagoras theorem is expressed in the form of trigonometry functions, it is said to be Pythagorean identity. There are majorly three identities:
- sin2 x + cos2 x = 1
- 1+tan2 x = sec2 x
- cosec2 x = 1 + cot2 x
These three identities are of great importance in Mathematics, as most of the trigonometry questions are prepared in exams based on them. Therefore, students should memorise these identities to solve such problems easily.
Sum and Difference Identities
- sin(x+y) = sin(x).cos(y)+cos(x).sin(y)
- sin(x–y) = sin(x).cos(y)–cos(x).sin(y)
- cos(x+y) = cosx.cosy–sinx.siny
- cos(x–y) = cosx.cosy+sinx.siny
- tan(x+y) = [tan(x)+tan(y)]/[1-tan(x)tan(y)]
- tan(x-y) = [tan(x)-tan(y)]/[1+tan(x)tan(y)]
Properties of Trigonometric Functions
Some of the common properties of trigonometric functions are discussed below:
Period refers to the length of one complete cycle of a trigonometric function, after which the function repeats.
- Sine (sin), Cosine (cos), Secant (sec), Cosecant (csc): Period = 2π
- Tangent (tan), Cotangent (cot): Period = π
Symmetry refers to the property that describes how the function behaves under reflection, translation, or rotation.
- Even Functions: f(−θ) = f(θ) (Cosine and Secant).
- Odd Functions: f(−θ) = −f(θ) (Sine, Tangent, Cosecant, Cotangent).