Some Basic Tangent Formulae

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

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.