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Tan Theta Formula

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Tangent is a function in Trigonometry which is a branch of mathematics that is concerned with specific functions of angles. It deals with the relationship between side lengths and angles of triangles. It is mainly used to find the unknown side lengths, angles of a right-angled triangle using trigonometric functions and formulas. There are six functions that are widely used in trigonometry. They are Sine (sin), Cosine (cos), Tangent (tan), Cotangent (cot), Secant(sec) and Cosecant (cosec). In this article, we are going to discuss the tan θ formula.

What is tan theta? 

tan θ is a commonly used trigonometric function along with other 5 functions. tan θ is also called as law of tangent. The tangent formula for a right-angled triangle can be defined as the ratio of the opposite side of a triangle to the adjacent side. It can also be represented as a ratio of the sine of the angle to the cosine of the angle.

Right-angled Triangle

tan θ = Opposite side/Adjacent side

where. 

θ is one of the acute angle.

Below are the tan values at different angles.

θ

0°

30°

45°

60°

90°

tan θ

0

1/√3

1

√3

∞

Important formulas for tan theta

  • tan(θ)=sin(θ)/cos(θ)
  • tan(θ)=1/cot(θ)
  • tan2(x)=sec2(x)-1
  • tan(-x)=-tan(x)
  • tan(90o-x)=cot(x)
  • tan(x+Ï€)=tan(x)
  • tan(Ï€-x)=-tan(x)
  • tan(x+y)= \frac{tan(x)+tan(y)}{1-tan(x).tan(y)}
  • tan(x-y)= \frac{tan(x)-tan(y)}{1+tan(x).tan(y)}
  • tan(2x)=\frac{2tan(x)}{1-tan^2(x)}
  • tan(3x)=\frac{3tan(x)-tan^3(x)}{1-3tan^2(x)}
  • tan(x/2)=\sqrt{\frac{1-cos(x)}{1+cos(x)}}

Sample Questions

Question 1: Find θ for a right-angled triangle if the length of the opposite side and adjacent side w.r.t θ are 3cm and 3√3 cm respectively.

Solution:

Given

Length of opposite side = 3 cm

Adjacent side length = 3√3 cm

From Tangent rule-

tan(θ) = Opposite side/Adjacent side

= 3/3√3

= 1/√3

tan 30° = 1/√3

tan(θ) = tan 30°

θ=30°

Question 2: Find tan θ for the given cot θ = 0.

Solution:

Given

cot θ = 0

the relation between tanθ and cotθ is inverse i.e.,

tan θ = 1/cot θ

=1/0

tan θ = ∞

Question 3: Find tan θ from the given sin θ = 1/2 and cos θ = √3/2.

Solution:

Given,

sin θ = 1/2

cos θ = √3/2

tan θ = sin θ / cos θ

= (1/2) / (√3/2)

= (1×2) / (2×√3)

tan θ = 1/√3

Question 4: Find tan x from the given sec x=2/5.

Solution:

Given

sec x = 2/5

we know that sec2 x – tan2 x = 1

From that tan2 x = sec2 x – 1

= (2/5)2-1

= (4/25)-1

= (4-25)/25

tan2(x) = -21/25

tan(x) = √(-21/25)

tan(x) = √(-21)/5

Question 5: Find the result of tan(60°+45°).

Solution:

Given

A=60° 

B=45°

We know that tan(A+B)= \frac{tan(A)+tan(B)}{1-tan(A).tan(B)}

tan(60°+45°)=\frac{tan(60°)+tan(45°)}{1-tan(60°).tan(45°)}

=(√3+1)/(1-√3×1)

=(√3+1)/(1-√3)

tan(60°+45°)=-3.732

Question 6: Calculate tan(x) where x=π-45°

Solution:

Given

tan(x)=tan(π-45°)

we know that tan(π-θ)=-tan(θ)

tan(π-45°)=-tan(45°)

tan(45°)=1

tan(π-45°)=-1


Last Updated : 18 Feb, 2024
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