Tangent Formula

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Sine, Cosine, and Tangent are the three basic functions of trigonometry through which trigonometric identities, trigonometry functions, and formulas are formed. The tangent is defined as the ratio of the length of the opposite side or perpendicular of a right angle to the angle and the length of the adjacent side. Tangent function in trigonometry is used to calculate the slope of a line between the origin and a point defining the intersection between hypotenuse and altitude of a right-angle triangle. In this article, we will discuss the tan formula, formula of a tangent.


What is Trigonometry?

It is the study of the relationships which involves angles, lengths, and heights of triangles given. It also relates to the different parts of circles as well as other geometrical figures. Trigonometry has many trigonometric ratios which are very fundamental in mathematics. It has many identities that are very useful for learning and deriving the many equations and formulas in science. There are many fields where these identities of trigonometry and formula of trigonometry are useful.


What is the Tangent Function?

Tangent is the ratio of the opposite side divided by the adjacent side in a right-angled triangle. In trigonometry, there are six possible ratios. A ratio is a comparison of two numbers i.e. sides of a triangle. The Greek letter,θ, will be used to represent the reference angle in the right triangle. These six ratios are useful in different ways to compare two sides of a right triangle.

Tan Formula is normally useful to calculate the angle of the right triangle. In a right triangle or the right-angled triangle, the tangent of an angle is a simple ratio of the length of the opposite side and the length of the adjacent side. Tangent is usually denoted as ‘tan’, but it is pronounced as a tangent. This function is useful to find out the length of a side of a triangle. It is possible when someone knows at least one side of the triangle and one of the acute angles.


 Derivation of the Tan Formula

As we know, Sine, Cosine, and Tangent are the three basic functions of trigonometry. Let us brief about all the three basic functions with the help of a right-angle triangle.

(Image to be added soon)


What is the Sine Function?

The Sine Function states that for a given right angle triangle, the Sin of angle ϴ is defined as the ratio of the length of the opposite side of a triangle to its hypotenuse.

Sin θ = Opposite side/ Hypotenuse


What is the Cosine Function?

The Cosine Function states that for a given right angle triangle, the Cosine of angle ϴ is defined as the ratio of the length of the adjacent side of a triangle to its hypotenuse.

Cos θ = Adjacent side / Hypotenuse


What is the Tangent Function?

The Tangent Function states that for a given right angle triangle, the Cosine of angle ϴ is defined as the ratio of the length of the opposite side of a triangle to the angle and the adjacent side.

Tan θ = Opposite side / Hypotenuse


Trigonometry Equations on the basis of Tangent Function (Tangent Formulas)

Various tangent formulas can be formulated through a tangent function in trigonometry. The basic formula of the tangent which is mostly used is to solve questions is,

Tan θ = Perpendicular/ Base or Tanθ = Sinθ/ Cosθ Or Tanθ = 1/Cotθ


Other Tangent Formulas Are

Tan (a+b) equals Tan (a) + Tan (b)/1- Tan (a) Tan (b)

Tan (90 + θ) = Cot θ

Tan (90 - θ) = - Cot θ

Tan (-ϴ) = Tanθ

Trigonometry Ratio Table of Different Angles

Angle

00

300

450

600

900

1800

2700

3600

Sin 

0

1/2

1/√2

√3/2

  1

0

-1

0

Cos 

1

√3/2

1/√2

½

  0

-1

0

1

Tan 

0

1/√3

1

√3

0

0

Cot 

√3

1

1/√3

  0

0

Cosec 

 2

√2

2/√3

1

-1

Sec 

1

2/√3

√2

   2

-1

1


Questions to be Solved 

Question 1

Calculate the tangent angle of a right triangle whose adjacent side and opposite sides are 8 cm and 6 cm respectively?

Solution

Given, Adjacent side i.e. base = 8 cm

Opposite side i.e. perpendicular = 6 cm

Also, the tangent formula is: Tan θ=perpendicular/base

Tan θ=6/8 = 0.75

Therefore, tan θ=0.75

Thus tangent value will be 0.75.


Question 2

Calculate the tangent angle of a right triangle whose adjacent side and opposite sides are 10 cm and 4 cm respectively?

Solution

Given, Adjacent side i.e. base = 10 cm

Opposite side i.e. perpendicular = 4 cm

Also, the tangent formula is: Tan θ=perpendicular/base

Tan θ=10/4 = 2.5

Therefore, tan θ= 2.5

Thus tangent value will be 2.5.

FAQ (Frequently Asked Questions)

1. What is Tan Equal to?

The tangent of x is defined to be its sine divided by its cosine: tan x = sin x/ cos x. The cotangent of x can be defined to be the cosine of x divided by the sine of x: cot x = cos x/ sin x.

2. What is the Symbol for a Tangent?

The ratio of the adjacent side of a right triangle to the hypotenuse is called the cosine and given the symbol cos. Now finally, the ratio of the opposite side to the adjacent side is known as the tangent and given the symbol tan.

3. Where is Tangent 1?

The exact value of arctan(1) is equal to π4. The tangent function is known to be positive in the first and third quadrants. Now to find the second solution, we need to add the reference angle from π to find the solution in the fourth quadrant.

4. What is the Symbol for Pi?

Pi is denoted by the symbol π and is defined as the ratio of the circumference of a circle to its diameter, pi, or in symbol form, π, which seems a simple enough concept. But it rather turns out to be an "irrational number," which means that its exact value is inherently unknowable.