Cot Tan Formula: Derivation & Relationship

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Cot Tan formula is a type of formula where Tan and Cot have inverse relations. The cot-tan formula indicates an inverse relationship between Cot θ and Tan θ. Trigonometry is a branch of maths which deals with the angles, lengths and sides of a triangle. There are six trigonometric ratios and these are the ratios of right angled triangle sides. The altitude of it consists of Tan θ and its base is Cot θ.

Table of Content

What is Cotangent
Tan Cot Formula
Relationship between Cot and Tan
Things to Remember
Previous Year Questions
Sample Questions

Trigonometry consists of 6 trigonometric functions which are actually the ratios of sides of a right-angled triangle. The six trigonometric identities are:

Sine
Cos or Cosecant
Tan Or tangent
Cosec Or cosecant
Sec or Secant
Cot or Cotangent

The cotangent (cot) of an angle can simply be expressed as the reciprocal of the tangent of the angle. Mathematically, the formula for the cotangent of an angle θ (in radians) is given as:

cot(θ) = 1/tan(θ)

Here,

tan(θ) is the tangent of the angle
θ is the angle in radians

Hence, the relationship between cotangent and tangent can be represented as the "Cotangent-Tangent Formula".

Key Terms: Tangent, Cotangent, Right angle triangle, Triangle, Pythagoras Theorem, Cosine, Secant, Cosecant, Tangent, Perpendicular, Hypotenuse, Integrals, Trigonometric Ratio, Trigonometry

Read More: Trigonometric Values

What is Cotangent

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Cotangent or Cot is one of the 6 trigonometric functions. It is usually referred to as "cot". Just like other trigonometric ratios, the cotangent formula is also defined as the ratio of the sides of a right-angled triangle. The cotangent formula is equal to the ratio of the base and perpendicular of a right-angled triangle.

The cotangent represents the ratio of the adjacent side to the opposite side of a right triangle. Given an angle θ in a right triangle, the cotangent of that angle, denoted as cot(θ), can be represented as:

cot(θ) = adjacent side/opposite side
or,
cot(θ) = 1/tan(θ)

where, tan(θ) is the tangent of the angle θ.

The cotangent is used to solve for unknown side lengths or angles in a right triangle, and it is one of the six basic trigonometric functions along with sine, cosine, tangent, secant, and cosecant.

Trigonometric Functions Detailed Video Explanation

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Relevant Concepts
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Trigonometry Table	Trigonometric Equations	Value of cos 60
Cosine Rule	Sine Function	Inverse Tan

Tan Cot Formula

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Cot and Tan of an angle θ represents an inverse relationship with each other.

Thus, Cot θ × Tan θ = 1

Simply, Tan θ can be defined as the ratio of the Opposite side to the Adjacent, wherein (θ) is among the acute angles.

Thus,

Tan Theta = Opposite/Adjacent
Cot Theta = Adjacent/Opposite

Cot Tan Theta Formula by Right Triangle

Cot Tan Theta Formula by Right Triangle

Cot Tan Formula Derivation

The cotangent-tangent formula can be derived from the very definitions of tangent and cotangent.

Consider an angle θ in a right triangle is given, where the tangent of the angle θ is simply the ratio of the opposite side to the adjacent side. Thus, it can be represented as:

tan(θ) = opposite side/adjacent side

And, the cotangent of the angle θ is simply the reciprocal of the tangent or the ratio of the adjacent side to the opposite side. It can be represented as:

cot(θ) = adjacent side/opposite side = 1/tan(θ)

Thus, the cotangent-tangent formula can be shown as:

→ cot(θ) = 1/tan(θ)

This formula expresses the relationship between the cotangent and tangent of an angle in a right triangle, and can be used to find the value of one if the value of the other is known.

Read Also: Tangents and Normals

Frequently Asked Question

Ques. Is cot the same as tan inverse of 1? (2 marks)

Ans. The cotangent function is simply a reciprocal of the tan inverse of 1.

which means, cot (x) = 1/tan (x)

The inverse of the tangent function can be expressed as tan^-1(x)

Thus, it can be said that, if y = tan(x), then, x = tan^-1(y)

Hence, cot(x) is not the same as tan^-1(x)

Relationship between Cot and Tan

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For understanding the relationship between the cotangent and Tangent of any angle θ , we have to understand the values of each.

Tangent of an angle θ = Altitude/Base

The cotangent of an angle θ = Base/ Altitude

Read Also: Trigonometry Table

As seen above the two functions are actually reciprocal of each other, thus Tangent could be written as,

Tangent of an angle θ = 1/Cot θ

or,

Cotangent of an angle θ = 1/ Tanθ

Due to this, we come across one more formula, I.e.,

Tanθ × Cot θ = (Altitude/Base) × (Base / Altitude) = 1 (the numerator and denominator cancels out)

Thus,

⇒ Tanθ × Cotθ = 1

Read Also:

Related Topics
Trigonometry Values	Sin Cos Formulas	Trigonometric Identities
Cos Square Theta Formula	Secant Square X Formula	Graphs of Inverse Trigonometric Functions

Things to Remember

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Cot-Tan is defined as the ratio of cot of an angle to the cos of the angle in a right-angled triangle.
The ratio of the opposite side to the adjacent side in the triangle in which (θ) is one of the acute angles, is known as Tan θ.
The cot-tan formula for the triangle is given as follows: Tan θ = Opposite side / Adjacent side and Cot θ = Adjacent side / Opposite side.
The cotangent of an angle θ is a trigonometric function denoting the reciprocal of the tangent of the angle, or the ratio of the length of the adjacent side to the length of the opposite side in a right triangle for that angle.
The cotangent is used in trigonometry to solve for unknown side lengths or angles in a right triangle, and it has several important properties, such as: cot(θ) = 1/tan(θ), cot(θ) = cot(π/2 - θ), cot(-θ) = -cot(θ) and cot(θ + π) = -cot(θ).

Previous Year Questions

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Sample Questions

Ques. What is sin theta in trigonometry? (2 marks)

Ans. Sin theta is the ratio of the perpendicular to the hypotenuse in the triangle. The formula of Sinθ is given below.

Sinθ = Opposite side/Hypotenuse

Ques. What is tangent? (2 marks)

Ans. The tangent is a trigonometric function which represents the ratio of the length of the opposite side to the length of the adjacent side in a right triangle for a given angle. Mathematically, the formula for the tangent of an angle θ (in radians) is given as:

tan(θ) = opposite side/adjacent side

Ques. What is Cotangent? (2 marks)

Ans. The cotangent is one of the main trigonometric ratios in trigonometry. So, it can be defined as the ratio between the two sides of a right-angled triangle.

This is known as Cotangent. Cotangent is generally referred to as cot.

Ques. Write the cot-tan formula. (2 marks)

Ans. The cot-tan formula is given below.

As seen from the previous derivation, the Cot and Tan of an angle θ present an inverse relationship.

Thus, Cotθ × Tanθ = 1

Ques. What do you mean by trigonometry? (3 marks)

Ans. Trigonometry is a branch of mathematics which deals with the lengths, sides and angles of a triangle. Various trigonometric identities are there which are related to each part of a triangle. The trigonometric identities are used to calculate an expression based on the angles, sides and length of the triangle.

Ques. What are trigonometric identities? (3 marks)

Ans. Trigonometric identities are defined as equations which can relate to various functions of trigonometry are known as trigonometric identities. There are mainly three types of trigonometric ratios i.e, sin, cos,and tan. The more three ratios are the reciprocals of other trigonometric ratios which are known as sec, cosec, and cot.

Ques. Write the uses of trigonometry. (3 marks)

Ans. Trigonometry is a vast part of mathematics. So, it has so many uses in particular fields. Let's discuss them one by one.

Trigonometry is used to set all four directions by using a device called a "compass".The compass shows us the right and straight direction.
It is used in the field of navigation.
Trigonometry is used to find the distance of the seashore from a particular point in the sea.
Trigonometry is also used in map making.

Ques. Demonstrate the cotangent function in terms of Sine and Cosine. (3 marks)

Ans. The cotangent function in terms of sine and cosine functions can be represented as,

cot θ = cos θ/sin θ

We are aware that, cot θ = adjacent side/opposite side

Dividing both the numerator and denominator with the given hypotenuse, we get:

⇒ cot θ = (adjacent side/hypotenuse) / (opposite side/hypotenuse)

We are also aware that, sin θ = opposite side/hypotenuse

thus,

cos θ = adjacent side/hypotenuse

Therefore, cot θ = cos θ/sin θ

Ques. Shown the Cotangent function in terms of cosecant function. (2 marks)

Ans. The Cotangent function in terms of Cosecant can be represented as:

cot θ = √(cosec² – 1)

Now, from the Pythagorean identities, we can say:

cosec² θ – cot² θ = 1

⇒ cot² θ = 1 – cosec² – 1

Hence,

cot θ = √(cosec² – 1)

Ques. Find the value of cot α, sin α = 1/3, and cos α = 2√2/3. (3 marks)

Ans. As per the given question, it can be said that,

sin α = 1/3 and cos α = 2√2/3

We are already aware that,

cot α = cos α/sin α

⇒ cot α = (2√2/3) / (1/3) = 2√2

Therefore, the value of cot α = 2√2.

Ques. Determine the following:
a) Value of cot θ if cosec θ = 25/24.
b) Value of cot β if sin β = 5/13. (5 marks)

Ans. a) As per the given question, cosec θ = 25/24

We are aware that, cot θ = √(cosec² – 1)

⇒ cot θ = √(25/24)² – 1

⇒ cot θ =√(625 – 576)/576 = √49/576

⇒ cot θ = 7/24

Thus, the value of cot θ = 7/24.

b) As per the given question, sin β = 5/13

We are aware that, sin²β + cos² β = 1

⇒ (5/13)²+ cos² β = 1

⇒ cos²β = 1 – (5/13)² = 1 – 25/169 = 144/169

⇒ cos β = √144/169 = 12/13

⇒ cot β = cos β/sin β

= (12/13) / (5/13)

⇒ cot β = 12/5

Therefore, the value of cot β = 12/5

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Cot Tan Formula: Derivation & Relationship

What is Cotangent

Tan Cot Formula

Cot Tan Formula Derivation

Frequently Asked Question

Relationship between Cot and Tan

Things to Remember

Previous Year Questions

Sample Questions

CBSE CLASS XII Related Questions

1.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

5.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

6.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :

Similar Mathematics Concepts

Comments

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Cot Tan Formula: Derivation & Relationship

What is Cotangent

Tan Cot Formula

Cot Tan Formula Derivation

Frequently Asked Question

Relationship between Cot and Tan

Things to Remember

Previous Year Questions

Sample Questions

CBSE CLASS XII Related Questions

1. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3.Find:Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

5.Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

6.At a birthday party, children are being served orange juice in conical cups, as shown in the figure. Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.On the basis of the above information, answer the following questions :

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

3.
Find:
Let \(A=[a_{ij}]\) be a \(2\times2\) matrix whose elements are given by \[ a_{ij}=\frac{(2i-j)^2}{3} \] Find the transpose matrix \(A'\).

4.
If \( xy = e^{x - y} \), then find \( \frac{dy}{dx} \).

5.
Differentiate \( \tan^{-1}\left( \frac{\sqrt{1 + x^2} + \sqrt{1 - x^2}}{\sqrt{1 + x^2} - \sqrt{1 - x^2}} \right) \) with respect to \( \cos^{-1}(x^2) \).

6.
At a birthday party, children are being served orange juice in conical cups, as shown in the figure.

Each cup is 15 cm deep and has a radius 5 cm. The juice is being poured into this cup at a rate of 0·1 cm3/s.
On the basis of the above information, answer the following questions :