What is trigonometric identity? Explain with examples.
An equation is called an identity when it is true for all values of the variables involved. Similarly, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angles involved.
Answer: Trigonometric ratios of sin, cos, tan, cot, sec, cosec are connected through a set of rules or identities.
There are a total of six trigonometric ratios. They are as follows:
sin θ = perpendicular / hypotenuse
tan θ = perpendicular / base
cot θ = base / perpendicular
cosec θ = hypotenuse / perpendicular
sec θ = hypotenuse / base
From the given ratios, we observe that sin and cosec are reciprocal of each other. These are called reciprocal identities
|sinθ = 1 / cosecθ ; cosec θ = 1/sinθ|
|cosθ = 1/secθ ; secθ = 1/cosθ|
|tanθ = 1 / cotθ ; cotθ= 1/tanθ|
|tanθ = sinθ/cosθ|
|cotθ = cosθ/sinθ|
We are deriving all the different identities of trigonometry with the help of the Pythagoras theorem.
Let us understand one of the identities through the Pythagoras theorem.
According to Pythagoras theorem,
(perpendicular)2 + (base)2 = (hypotenuse)2 --------(1)
let us write sin A = perpendicular / hypotenuse ------(2)
and cos A = base / hypotenuse -------(3)
On squaring and adding equation (2) and (3), we get
sin2 A + cos2 A = (perpendicular / hypotenuse)2 + (base / hypotenuse)2
sin2 A + cos2 A = (perpendicular 2 + base2) / hypotenuse2
sin2 A + cos2 A = hypotenuse2/ hypotenuse2 [from equation (1) ]
sin2 A + cos2 A = 1
Hence , sin2 A + cos2 A = 1 is an identity of trigonometry.
Similarly we can also derive other identities by using pythagoras theorem.
|sin2θ + cos2θ = 1 (⇒ cos2θ = 1 - sin2θ; sin2θ = 1 - cos2θ)|
|1 + tan2θ = sec2θ (⇒ tan2θ = sec2θ - 1 ; sec2θ - tan2θ = 1)|
|cosec2θ - cot2θ = 1 ( ⇒ 1 + cot2θ = cosec2θ ; cot2θ = cosec2θ - 1)|
Example: If sin θ is 4 / 5, then find the value for cos θ.
Solution: Using trigonometric identity
sin2θ + cos2θ = 1
⇒ cos2θ = 1 - sin2θ
⇒ cos2θ = 1 - (16/25)
⇒ cos2θ = 9 / 25
⇒ cos θ = 3 /5 [ taking square root of 9 / 25 ]
There are other identities like opposite angle identities, half-angle identities, cofunction identities some of it are listed below
|Opposite Angle Identities (Even Odd Identities)|
|cos(−θ) = cos(θ), sec(−θ) = sec(θ)|
|tan(−θ) = −tan(θ), cot(−θ) = - tan(θ)|
|sin(−θ) = −sin(θ), cosec(−θ) = - cosec(θ)|
|Double Angle Identities|
|sin2θ = 2 sinθcosθ = 2 tanθ/ (1 + tan2θ)|
|cos2θ = cos2θ - sin2θ (= (1- tan2θ)/ (1 + tan2θ)|
|tan2θ = 2tanθ/ 1- tan2θ|
|sin(π/2 - θ) = cosθ ; cos (π/2 - θ) = sinθ ;|
|tan(π/2 - θ) = cotθ ; cot(π/2 - θ) = tanθ ;|
|sec(π/2 - θ) = cosecθ ; cosec (π/2 - θ) = secθ ;|