Secant Is The Reciprocal Of

Reciprocal Identities

Reciprocal Identities are the reciprocals of the six chief trigonometric functions, namely sine, cosine, tangent, cotangent, secant, cosecant. The important thing to annotation is that reciprocal identities are not the same every bit the inverse trigonometric functions. Every central trigonometric part is a reciprocal of some other trigonometric function. For example, cosecant is the reciprocal identity of the sine function.

In this article, we volition determine the reciprocal identities, prove the reciprocal identities and find the relationship between them with the help of solved examples.

1.	What are Reciprocal Identities?
2.	Reciprocal Identities Formulas
3.	Proof of Reciprocal Identities
4.	Relationship of Reciprocal Identities
5.	FAQs on Reciprocal Identities

What are Reciprocal Identities?

The reciprocals of the six central trigonometric functions (sine, cosine, tangent, secant, cosecant, cotangent) are called reciprocal identities. The reciprocal identities are of import trigonometric identities that are used to solve various bug in trigonometry. Each trigonometric role is a reciprocal of another trigonometric function. The sine function is the reciprocal of the cosecant role and vice-versa; the cosine part is the reciprocal of the secant part and vice-versa; cotangent office is the reciprocal of the tangent function and vice-versa.

Reciprocal Identities Formulas

Reciprocal identities are applied in various trigonometry problems to simplify the calculations. The formulas of the six primary reciprocal identities are:

sin ten = 1/cosec x
cos x = 1/sec 10
tan x = i/cot x
cot x = 1/tan x
sec 10 = 1/cos x
cosec ten = 1/sin x

Reciprocal Identities

Proof of Reciprocal Identities

Now, that we know the reciprocal identities of trigonometry, allow us now bear witness each one of them using the definition of the basic trigonometric functions. Beginning, we volition derive the reciprocal identity of the sine role. Consider a right-angled triangle ABC with a right angle at C.

reciprocal identities proof

We know that sin θ = Perpendicular/Hypotenuse = c/a and cosec θ = Hypotenuse/Perpendicular = a/c ⇒ sin θ is the reciprocal of cosec θ and cosec θ is the reciprocal of sin θ. Similarly, we will prove other reciprocal identities. cos θ = Base/Hypotenuse = b/a and cosec θ = Hypotenuse/Base of operations = a/b ⇒ cos θ is the reciprocal of sec θ and sec θ is the reciprocal of cos θ. tan θ = sin θ/cos θ and cot θ = cos θ/sin θ ⇒ tan θ is the reciprocal of cot θ and cot θ is the reciprocal of tan θ. Hence, nosotros have

sin θ is the reciprocal of cosec θ
cosec θ is the reciprocal of sin θ
cos θ is the reciprocal of sec θ
sec θ is the reciprocal of cos θ
tan θ is the reciprocal of cot θ
cot θ is the reciprocal of tan θ

Relationship of Reciprocal Identities

As we know that the product of a number and its reciprocal is always equal to one, we have established similar relationships between the reciprocal identities. The product of a trigonometric role and its reciprocal is equal to one. Hence, we have

sin θ × cosec θ = one
cos θ × sec θ = 1
tan θ × cot θ = 1

The above equations plant a relationship between the reciprocal identities of trigonometry for any bending θ and show that the product of a trigonometric role and its reciprocal is equal to one.

Important Notes on Reciprocal Identities

sin θ is the reciprocal of cosec θ
cosec θ is the reciprocal of sin θ
cos θ is the reciprocal of sec θ
sec θ is the reciprocal of cos θ
tan θ is the reciprocal of cot θ
cot θ is the reciprocal of tan θ

Related Topics on Reciprocal Identities

Changed Trigonometric Formulas
Trigonometric Formulas
Trigonometric Table
Trigonometry

Reciprocal Identities Examples

Case 1: Simplify the expression [cot 10(sin x + tan x)/(cosec 10 + cot ten)] using reciprocal identities.

Solution: We will write the expression [cot x(sin x + tan 10)/(cosec x + cot ten)] in terms sin x and cos x and use the reciprocal identity cosec x = 1/sin x

[cot ten(sin x + tan x)/(cosec x + cot x)] = [(cos ten/sin x)(sin 10 + (sin x/cos x))/((1/sin x) + (cos x/sin ten))]

= (cos 10 + 1)(sin x)/(ane + cos x)

= sin ten

Answer: [cot x(sin x + tan x)/(cosec x + cot x)] = sin x
Example two: Determine the value of sec x if cos x = 3/7 using reciprocal identity.

Solution: We know the reciprocal identity sec x = ane/cos x

And then, if cos x = three/7, and then sec 10 = one/cos 10 = 1/(3/seven) = 7/3

Answer: sec x = 7/3

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FAQs on Reciprocal Identities

What are Reciprocal Identities in trigonometry?

The reciprocals of the six key trigonometric functions (sine, cosine, tangent, secant, cosecant, cotangent) are called reciprocal identities. The sine function is the reciprocal of the cosecant part and vice-versa; the cosine role is the reciprocal of the secant function and vice-versa; cotangent part is the reciprocal of the tangent function and vice-versa.

What are the Half dozen Reciprocal Identities?

The formulas of the six main reciprocal identities are:

sin x = 1/cosec 10
cos x = one/sec ten
tan x = i/cot x
cot 10 = one/tan x
sec 10 = ane/cos ten
cosec x = 1/sin x

What is the Reciprocal Identity of Cos x?

The reciprocal identity of cos x is sec ten considering cos x = 1/sec x. The secant function is the reciprocal of the cosine role.

When to Use Reciprocal Identities?

The reciprocal identities are very useful when solving trigonometric equations. If yous find a mode to multiply each side of an equation past a trigonometric function'southward reciprocal, yous may be able to reduce some part of the equation and simplify information technology.

What is the Reciprocal Identity of Sin x?

The reciprocal identity of sin x is cosec x considering sin 10 = 1/cosec x. The cosecant role is the reciprocal of the sine function.

What is the Difference Betwixt Quotient and Reciprocal Identities?

In trigonometry, quotient identities refer to trigonometric identities that are divided by each other whereas reciprocal identities are ones that are the multiplicative inverses of the trigonometric functions.