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 (1st quadrant)
- tan (π – θ) = – tan θ (2nd quadrant)
- tan (π + θ) = tan θ (3rd quadrant)
- tan (2π – θ) = – tan θ (4th 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:
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.
Table of Content
- Trigonometric Ratios
- Tangent Formula
- Some Basic Tangent Formulae
- Examples on Tangent Formulas
- FAQs on Tangent Formula