Tangent Formulas

Tangent Function is among the six basic trigonometric functions and is calculated by taking the ratio of the perpendicular side and the hypotenuse side of the right-angle triangle.

In this article, we will learn about Trigonometric ratios, Tangent formulas, related examples, and others in detail.

Trigonometric ratios are ratios of sides in a triangle and there are six trigonometric ratios. In a right-angle triangle, the six trigonometric ratios are defined as:

1. sin θ = (Opposite Side/Hypotenuse
2. cos θ = Adjacent Side/Hypotenuse
3. tan θ = Opposite side/adjacent side
4. cosec θ = 1/sin θ = Hypotenuse/Opposite Side
5. sec θ = 1/cos θ = Hypotenuse/Adjacent Side
6. cot θ = 1/tan θ = Adjacent Side/Opposite Side

Tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side to the given angle. We write a tangent function as “tan”.

Let us consider a right-angled triangle ABC and one of its acute angles is “θ”. An opposite side is the side that is opposite to the angle “θ” and the adjacent side is the side that is adjacent to the angle “θ”.

Now, the tangent formula for the given angle “θ” is,

Tangent Function in Quadrants

The tangent function is positive in the first and third quadrants and negative in the second and fourth quadrants.

• tan (2π + θ) = tan θ (1^st quadrant)
• tan (π – θ) = – tan θ (2^nd quadrant)
• tan (π + θ) = tan θ (3^rd quadrant)
• tan (2π – θ) = – tan θ (4^th quadrant)

Tangent Function as a Negative Function

The tangent function is a negative function since the tangent of a negative angle is the negative of a tangent positive angle.

tan (-θ) = – tan θ

Tangent Function in Terms of Sine and Cosine Function

Tangent function in terms of sine and cosine functions can be written as,

tan θ = sin θ/cos θ

We know that, tan θ = Opposite side/Adjacent side

Now, divide both the numerator and denominator with hypotenuse

tan θ = (Opposite side/Hypotenuse)/(Adjacent side/Hypotenuse)

We know that, sin θ = opposite side/hypotenuse

cos θ = adjacent side/hypotenuse

Hence, tan θ = sin θ/cos θ

Tangent Function in Terms of Sine Function

Tangent function in terms of the sine function can be written as,

tan θ = sin θ/(√1 – sin² θ)

We know that,

tan θ = sin θ/cos θ

From the Pythagorean identities, we have,

sin² θ + cos² θ = 1

cos² θ = 1 – sin² θ

cos θ = √(1 – sin² θ)

Hence, tan θ =  sin θ/(√1 – sin² θ)

Tangent Function in Terms of Cosine Function

Tangent function in terms of the cosine function can be written as,

tan θ = (√1 -cos² θ)/cos θ

We know that,

tan θ = sin θ/cos θ

From the Pythagorean identities, we have,

sin² θ + cos² θ = 1

sin² θ = 1 – cos² θ

sin θ = √(1 – cos² θ)

Hence, tan θ =  (√1 – cos² θ)/cos θ

Tangent Function in Terms of Cotangent Function

Tangent function in terms of the cotangent function can be written as,

tan θ = 1/cot θ

or

tan θ = cot (90° – θ) (or) cot (π/2 – θ)

Tangent Function in Terms of Cosecant Function

Tangent function in terms of the cosecant function can be written as,

tan θ = 1/√(cosec² θ – 1)

From the Pythagorean identities, we have,

cosec² θ – cot² θ = 1

cot² θ = cosec² θ – 1

cot θ = √(cosec² θ – 1)

We know that,

tan θ = 1/cot θ

Hence, tan θ = 1/√(cosec² θ – 1)

Tangent Function in Terms of Secant Function

Tangent function in terms of the secant function can be written as,

tan θ = √sec² θ – 1

From the Pythagorean identities, we have,

sec² θ – tan² θ = 1

tan θ = sec² θ – 1

Hence, tan θ = √(sec² θ – 1)

Tangent Function in Terms of Double Angle

Tangent function for a double angle is,

tan 2θ = (2 tan θ)/(1 – tan² θ)

Tangent Function in Terms of Triple Angle

Tangent function for a triple angle is,

 tan 3θ = (3 tan θ – tan³θ) / (1 – 3 tan²θ)

Tangent Function in Terms of Half-Angle

Tangent function for a half-angle is,

tan (θ/2) = ± √[ (1 – cos θ) / (1 + cos θ) ]

tan (θ/2) = (1 – cos θ) / ( sin θ)

Tangent Function in Terms of Addition and Subtraction of Two Angles

Sum and difference formulas for a tangent function are,

tan (A + B) = (tan A + tan B)/(1 – tan A tan B)

tan (A – B) = (tan A – tan B)/(1 + tan A tan B)

Article Related to Tangent Formula:

• Trigonometric Identites
• Trigonometric Function
• Trigonometric Table
• Sin Cos Tan values

Example 1: Find the value of tan θ if sin θ = 2/5 and θ is the first quadrant angle.

Solution:

Given,

• sin θ = 2/5

From the Pythagorean identities we have,

sin² θ + cos² θ = 1

cos² θ = 1 – sin² θ = 1 – (2/5)²

cos² θ = 1 – (4/5) = 21/25

cos θ = ±√21/5

Since θ is the first quadrant angle, cos θ is positive.

cos θ = √21/5

We know that,

 tan θ = sin θ/cos θ

= (2/5)/(√21/5) = 2/√21

tan θ = 2√21 /21

So, value of tan θ when sin θ = 2/5 and θ is in first quadrant is (2√21) /(21)

Example 2: Find the value of tan x if sec x = 13/12 and x is the fourth quadrant angle.

Solution:

Given, sec x = 13/12

From the Pythagorean identities, we have,

sec² x – tan² x = 1

tan² x = sec² x – 1= (13/12)² – 1

tan² x = (169/144) – 1= 25/144

tan x = ± 5/12

Since x is the fourth quadrant angle, tan x is negative.

tan x = – 5/12

Hence, tan x = – 5/12

Example 3: If tan X = 2/3 and tan Y = 1/2, then what is the value of tan (X + Y)?

Solution:

Given,

tan X = 2/3 and tan Y = 1/2

We know that,

tan (X + Y) = (tan X + tan Y)/(1 – tan X tan Y)

tan (X + Y) = [(2/3) + (1/2)]/[1 – (2/3)×(1/2)]

= (7/6)/(2/3) = 7/4

Hence, tan (X + Y) = 7/4

Example 4: Calculate the tangent function if the adjacent and opposite sides of a right-angled triangle are 4 cm and 7 cm, respectively.

Solution:

Given,

Adjacent side = 4 cm

Opposite side = 7 cm

We know that,

tan θ = Opposite side/Adjacent side

tan θ = 7/4 = 1.75

Hence, tan θ = 1.75

Example 5: A man is looking at a clock tower at a 60° angle to the top of the tower, whose height is 100 m. What is the distance between the man and the foot of the tower?

Solution:

Given,

Height of tower = 100 m and θ = 60°

Let distance between man and foot of tower = d

We have,

tan θ = Opposite side/Adjacent side

tan 60° = 100/d

√3 = 100/d [Since, tan  60° = √3]

d = 100/√3

Therefore, distance between the man and the foot of tower is 100/√3

Example 6: Find the value of tan θ if sin θ = 7/25 and sec θ = 25/24.

Solution:

Given,

sin θ = 7/25

sec θ = 25/24

We know that,

sec θ = 1/cos θ 

25/24 = 1/cos θ cos θ = 24/25

We have,

tan θ = sin θ/cos θ

= (7/25)/(24/25)

= 7/24

Hence, tan θ = 7/24

Example 7: Find the value of tan θ if cosec θ = 5/3, and θ is the first quadrant angle.

Solution:

Given, cosec θ = 5/3

From the Pythagorean identities, we have,

cosec² θ – cot² θ = 1

cot² θ = cosec² θ – 1

cot θ = (5/3)² – 1 = (25/9) – 1 = 16/9

cot θ = ±√16/9 = ± 4/3

Since θ is the first quadrant angle, both cotangent and tangent functions are positive.

cot θ = 4/3

We know that,

cot θ = 1/tan θ 

4/3 = 1/tan θ

tan θ = 3/4

Hence, tan θ = 3/4

Example 8: Find tan 3θ if sin θ = 3/7 and θ is the first quadrant angle.

Solution:

Given, sin θ = 12/13

From the Pythagorean identities we have,

sin²θ + cos²θ = 1

cos²θ = 1 – sin²θ = 1 – (12/13)²

cos2 θ = 1 – (144/169) = 25/169

cos θ = ±√25/169 = ±5/13

Since θ is the first quadrant angle, cos θ is positive.

cos θ = 5/13

We know that,

tan θ = sin θ/cos θ

= (12/25)/(5/13) = 12/5

Hence, tan θ = 12/5

Now, We know that ,

tan 3θ = (3 tan θ – tan3θ) / (1 – 3 tan2θ)

tan 3θ = 3 × (12/5)

What is the rule for tangent?

Rule for tangent states that tangent is equal to the ratio of the perpendicular and base and is the resiprocal of cotenget function and is represented as, tan x = 1/cot x.

How to calculate the cotangent?

Cotangent is calculated using the formula, cot A = Base/Perpendicular

Is tangent equal to cotangent?

No, cotangent is not equal to tangent, as they are reciprocal of each other.

What is the formula for tangent?

The formula for tangent is opposite over adjacent.