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Reciprocal identities are the reciprocals of the three standard trigonometric functions, namely sine, cosine, and also tangent. In trigonometry, reciprocal identities are sometimes called inverse identities. Mutual identities space inverse sine, cosine, and tangent attributes written as “arc” prefixes such as arcsine, arccosine, and also arctan. Because that instance, functions like sin^-1 (x) and cos^-1 (x) space inverse identities. Either notation is correct and acceptable.


Learning reciprocal identities requires you come be acquainted with the formulas of the simple trigonometric identities of sine, cosine, and also tangent. These an essential identities room inverses that the functions. They action as features reversing the location of the numerator and also denominator. In this case, the political parties of the triangle.

The reciprocal for sine function is the cosecant function.


sin (θ) = 1 / csc (θ)

The mutual for cosine duty is secant function.

cos (θ) = 1 / sec (θ)

The reciprocal for tangent function is cotangent function.


tan (θ) = 1 / cot (θ)

The reciprocal for cosecant function is sine function.

csc (θ) = 1 / sin (θ)

The mutual for secant role is cosine function.

sec (θ) = 1 / cos (θ)

The reciprocal for cotangent role is tangent function.

cot (θ) = 1 / tan (θ)

One crucial thing to keep in mind is that all definitions of secant, cosecant, and cotangent involve separating something that could be zero. For example if cos (x) = 0 climate sec(x) is no defined because we’d be dividing by zero. The is why features like secant, cosecant, and also cotangent have holes or gaps in your domains.


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Reciprocal Identities Definition


Secant

The secant function is the reciprocal of the cosine. That is the hypotenuse proportion to the side nearby to a provided angle in a appropriate triangle. View the figure below to recognize the ide more.


cos (A) = surrounding / hypotenuse = b / c

sec (A) = hypotenuse / surrounding = c / b


Cosecant

The cosecant is the mutual of the sine function. The cosecant is the proportion of the hypotenuse of the ideal triangle to the next opposite a provided angle.

sin (A) = the opposite / hypotenuse = a / c

csc (A) = hypotenuse / the opposite = c / a

Cotangent

The cotangent role is the reciprocal of the tangent. It is the proportion of the nearby side to the opposite next in a provided right triangle.

tan (A) = the contrary / surrounding = a / b

cot (A) = surrounding / opposite = b / a

Other basic Trigonometric Identities applying the reciprocal Identities

Here space some different important identities or recipe that deserve to be used along with the reciprocal identities once simplifying or verifying identities.

Pythagorean Identities

sin2 (θ) + cos2 (θ) = 1

tan2 (θ) + 1 = sec2 (θ)

cot2 (θ) + 1 = csc2 (θ)

Quotient Identities

tan (θ) = sin (θ) / cos (θ)

cot (θ) = cos (θ) / sin (θ)

Steps for Proving Expressions entailing Reciprocal Identities

When working with trigonometric expressions, the is often desirable to convert one form to one equivalent function that may be an ext useful. Listed below are some of the techniques used come establish details identities. Remember the there is no fixed procedure that works in the proofs for all identities. Nevertheless, you can constantly proceed with particular steps that will aid in countless forms.


Start manipulating the more facility side and transform that into much more simple trigonometric expressions. Often, you solve the side wherein you can malfunction equations into uncomplicated trigonometric features like sine, cosine, and also tangent. Try algebraic work such together multiplying, factoring, combining and also splitting single fractions, and so on.If other steps fail, refer each function in terms of sine and also cosine functions and also then perform proper algebraic operations,At every step, keep the other side that the identity in mind. It often reveals what one need to do to gain there.

See the instances illustrated below to completely understand the mutual identities and also how lock are related to other an essential identities.


Example 1: recognize Trigonometric function Values using the mutual Identities

Find the worth of the adhering to expressions using the reciprocal identities.

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sin (x) = 3 / 7, csc (x) = ?cos (x) = √3 / 2, sec (x) = ?tan (x) = 3, cot (x) = ?sec (x) = π / 5, cos (x) = ?csc (x) = 0.5, sin (x) = ?cot (x) =√2 / 2, tan (x) = ?

Solution

Given the 6 trigonometric identities, get the reciprocals the each using the mutual identities pointed out earlier.