Opposite Of Tangent

In Trigonometry, we have six important trigonometric functions. The main three trigonometric functions equations are sine, cosine, tangent. The opposite three trigonometric functions are cosecant, secant, cotangent.

So, Opposite of tangent is Cotangent.

Cotangent is shortly called as ctg.

Cotangent is also referred as reciprocal of tangent trigonometric function.

The cotangent of angle ? in a right-angle triangle is the ratio of adjacent side and opposite side.

Cotangent is used to find unknown angle and unknown side.

The notation of Cotangent is cot ? (or) ctg ?.

Let us see some important formulas of opposite of tangent.
Formulas of Opposite of Tangent:

Formula for cotangent in a right triangle:

Cot ? = `("adjacent side")/("opposite side") `

Pythagorean identities for ctg:

1 + cot2 ? = csc2 ?

We can also rewritten the above one.

Cot2 ? = csc2 ? – 1

Derivative of ctg:

`d/dx` cot x   = - csc2 x

Integral of ctg:

`int`cot x dx = ln(sin x) + C

The cotangent value table for standard angles is shown below.

Angle/Function


   0°


30° (or) `pi/6`


45° (or) `pi/4`


60° (or) `pi/3`


90° (or) `pi/2`

    Cot ?


   8


  `sqrt(3)`


     1


  `1/sqrt(3)`


  0



Negative argument of ctg:

Cot(-?) = - cot(?)

Double angle of ctg:

Cot(2?) =` (cot^2 theta-1)/(2cot theta)` = `1/2` (cot ? – tan ?)

Triple angle of ctg:

Cot(3?) = `(cot^3 theta-3 cot theta)/(3 cot^2 theta-1)`

Half angle of ctg:

Cot(`theta/2`) = cot ? + csc ?

Cot(`theta/2`) = `(sin theta)/(1-cos theta)`

Cot(`theta/2`) = `sqrt((1+cos theta)/(1-cos theta))`

Relation between cotangent and inverse cotangent:

Cot(cot-1(?)) = ?

Simple Relation between cotangent and other trigonometric functions:

Cot(?) = tan(`pi/2` – ?)

Cot(?) = `(cos theta)/(sin theta)`

Let us see sample problem of opposite of tangent.
Example Problem - Opposite of Tangent:

My forthcoming post is on Definition of Continuity, and  Precalculus Problems will give you more understanding about Algebra

Example Problem 1:

Find the angle using cotangent if adjacent side is 5 cm and opposite side is 12 cm in a right triangle. (note: ? be angle)

Solution:

As we know that,

Cot ? = `("adjacent side")/("opposite side") `

cot ? = `5/12`

cot ? = 0.4167

Take inverse cotangent both sides. we get,

? = cot -1(0.4167)        

? = 67.4°

Therefore, angle of right triangle is 67.4°.

Example Problem 2:

Find the value of cot 45° using sin 45° = `1/sqrt(2)` and cos 45° = `1/sqrt(2)` .

Solution:

As we know that,

Cot ? = `(cos theta)/(sin theta)`

So, cot 45° = `cos 45/sin 45`

cot 45° = `(1/sqrt(2))/(1/sqrt(2))`

Cot 45° = 1.