Cot Tan Formula

Cot Tan formula

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. There are six trigonometric ratios namely, sine, cosine, tangent, cotangent, cosecant and secant.

The tangent of an angle is defined as the ratio of the length of the opposite side to the adjacent side in a right triangle, making it an important function for calculating slopes and angles of elevation and depression. Conversely, the cotangent is the reciprocal of the tangent, representing the ratio of the adjacent side to the opposite side. Learn more about Tan and Cot formulas in this article by Extramarks.

What are Tan and Cot?

Tangent and cotangent are trigonometric functions that relate the angles of a right triangle to the ratios of its sides. They are fundamental in trigonometry and have important applications in various fields, including mathematics, physics, engineering, and computer science.

Tangent(Tan)

The tangent of an angle in a right triangle is defined as the ratio of the length of the perpendicular (side opposite to the assumed angle) and base (side adjacent to assumed angles)

tan(θ) = opposite/adjacent = Perpendicular/Base = p/b

Cotantengent(Cot)​

The cotangent of an angle is the reciprocal of the tangent. It is the ratio of the length of the adjacent side to the length of the opposite side.

cot(θ) = adjacent/opposite = base/perpendicular = b/p

Cot Tan Formula

Cot and Tanhn are two trigonometric function that are reciprocal of each other. Hence, the product of cot and tan is 1.

The “Tan Cot Formula” typically refers to the fundamental relationships and identities involving the tangent (tan) and cotangent (cot) functions in trigonometry. Here are some key formulas and identities that involve tangent and cotangent:

Pythagorean Identity

The Pythagorean identities relate the tangent and cotangent functions to the secant (sec) and cosecant (csc) functions:

\[ 1 + \tan^2(\theta) = \sec^2(\theta) \]

\[ 1 + \cot^2(\theta) = \csc^2(\theta) \]

Angle Sum and Difference Formulas

For two angles \( \alpha \) and \( \beta \):

Tangent:

\[ \tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 – \tan(\alpha)\tan(\beta)} \]

\[ \tan(\alpha – \beta) = \frac{\tan(\alpha) – \tan(\beta)}{1 + \tan(\alpha)\tan(\beta)} \]

Cotangent:

\[ \cot(\alpha + \beta) = \frac{\cot(\alpha)\cot(\beta) – 1}{\cot(\alpha) + \cot(\beta)} \]

\[ \cot(\alpha – \beta) = \frac{\cot(\alpha)\cot(\beta) + 1}{\cot(\beta) – \cot(\alpha)} \]

Co-Function Identities

These identities relate the trigonometric functions of complementary angles:

\[ \tan(90^\circ – \theta) = \cot(\theta) \]

\[ \cot(90^\circ – \theta) = \tan(\theta) \]

Periodicity

Both the tangent and cotangent functions have periodic properties:

\[ \tan(\theta + \pi) = \tan(\theta) \]

\[ \cot(\theta + \pi) = \cot(\theta) \]

Symmetry and Other Properties

\[ \tan(-\theta) = -\tan(\theta) \]

\[ \cot(-\theta) = -\cot(\theta) \]

Relationship between Cot θ and Tan θ

We know that the formula for cot and tan is given below:

Tan θ = Opposite Side / Adjacent Side

Cot θ = Adjacent Side/Opposite Side

Hence, cot θ x tan θ = 1

Derivation of cot tan formula

To derive the cot tan formula, we start by recalling their definitions in terms of sine (sin) and cosine (cos).

The tangent of an angle \(\theta\) is defined as:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]

The cotangent of an angle \(\theta\) is defined as:
\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]

Now, to find the relationship between cotangent and tangent, we need to express one in terms of the other.

Let’s start with the definition of cotangent:
\[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \]

We can rewrite this as:
\[ \cot(\theta) = \frac{1}{\frac{\sin(\theta)}{\cos(\theta)}} \]

Note that \(\frac{\sin(\theta)}{\cos(\theta)}\) is the definition of tangent:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]

Substituting \(\tan(\theta)\) into the equation for \(\cot(\theta)\):
\[ \cot(\theta) = \frac{1}{\tan(\theta)} \]

Therefore, the relationship between cotangent and tangent is:
\[ \cot(\theta) = \frac{1}{\tan(\theta)} \]

This relationship shows that cotangent is the reciprocal of tangent.

Solved Examples

Example 1: If \(\tan(\theta) = 2\), find \(\cot(\theta)\).

Solution:

Using the formula \(\cot(\theta) = \frac{1}{\tan(\theta)}\):
\[ \cot(\theta) = \frac{1}{2} \]

So, \(\cot(\theta) = 0.5\).

Example 2: If \(\cot(\theta) = 4\), find \(\tan(\theta)\).

Solution:

Using the formula \(\cot(\theta) = \frac{1}{\tan(\theta)}\), we can solve for \(\tan(\theta)\):
\[ \tan(\theta) = \frac{1}{\cot(\theta)} = \frac{1}{4} \]

So, \(\tan(\theta) = 0.25\).

Example 3: Verify the relationship \(\cot(\theta) = \frac{1}{\tan(\theta)}\) for \(\theta = 45^\circ\).

Solution:

For \(\theta = 45^\circ\):

\[ \tan(45^\circ) = 1 \]

Using the formula \(\cot(\theta) = \frac{1}{\tan(\theta)}\):
\[ \cot(45^\circ) = \frac{1}{1} = 1 \]

We know that:
\[ \cot(45^\circ) = 1 \]

Since both \(\tan(45^\circ)\) and \(\cot(45^\circ)\) are equal to 1, the relationship \(\cot(\theta) = \frac{1}{\tan(\theta)}\) is verified for \(\theta = 45^\circ\).