Tan Theta Formula, Definition , Solved Examples

Tan Theta Formula: The tangent function, part of Trigonometry, a mathematical branch focusing on angle-related functions, explores the correlation between angles and side lengths within triangles. It primarily aids in determining unknown angles or side lengths within right-angled triangles using specific trigonometric functions and formulas. Trigonometry encompasses six widely utilized functions: Sine (sin), Cosine (cos), Tangent (tan), Cotangent (cot), Secant (sec), and Cosecant (cosec). This discussion centers on the formula for tan θ.

Tan Theta Formula

Tan θ, one of the six primary trigonometric functions, is also known as the tangent law. This function, in the context of a right-angled triangle, is defined as the ratio between the side opposite the angle and the adjacent side. Alternatively, it can be expressed as the ratio of the sine of an angle to its cosine.

tan θ = Opposite side/Adjacent side

where. 

θ is one of the acute angles.

θ	0°	30°	45°	60°	90°
tan θ	0	1/√3	1	√3	∞

Important Tan Theta Formula

Below are some Important Tan Theta Formulas.

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+π)=tan(x)

tan(π-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)}}

Tan Theta Formula Solved Examples

Example 1: Finding θ in a Right-Angled Triangle 

Solution: Given: Opposite side length = 3 cm 

Adjacent side length = 3√3  ​ cm

 Using the tangent rule: 

tan(θ)=  Opposite side ​ /  Adjacent side 

= 3 / 3√3  ​

 =  1 / √3  ​ 

tan ⁡ ( 3 0 ∘ ) = 1 / √3 

tan(30 ∘ )=1 / √3 

Therefore, θ=30 ∘ . 

Example 2: Determining tanθ from cotθ=0 

Solution: Given: cotθ=0 

As the relationship between tanθ and cotθ is inverse: 

tanθ= 1/  cotθ ​ = 1/ 0

tanθ=∞ 

Example 3: Finding tan ⁡  tanθ from sinθ= 1/2  ​ and cosθ= √3/2. ​ ​ 

Solution: Given: sinθ= 1/2 

cosθ= √3/2. ​  

Using the relationship 

tanθ= sinθ / cosθ​ : 

tanθ= (1/2 ) /​ ​ (√3/2) ​ ​ = 1 / √3 

Example 4: Determine tanx from secx= 2/5 ​ 

Solution: 

Given: secx= 2 /5 ​  ​

Using the identity sec 2 x−tan 2 x=1 and 

tan 2 x=sec 2 x−1: 

tan 2 x=( 2/5  ​ ) 2 −1= 4/25 ​ −1 = 4−25 ​/ 25  =− 21 /25 ​ 

tan(x) = √(-21/25)

tan(x) = √(-21)/5

Example 5: Determine tanx from secx= 1/5 ​

Solution:

Given: secx= 1 /5 ​  ​

Using the identity sec ² x−tan ² x=1 and

tan ² x=sec ² x−1:

tan ² x=( 1/5  ​ ) 2 −1= 1/25 ​ −1 = (1−25) ​/ 25  =− 24 /25 ​

tan(x) = √(-24/25)

tan(x) = √(-24)/25

Example 6: Finding tan ⁡  tanθ from sinθ= 1/ √2 ​ and cosθ= √3/2. ​ ​ 

Solution: Given: sinθ= 1/ √2 ​

cosθ= √3/2. ​  

Using the relationship 

tanθ= sinθ / cosθ​ : 

tanθ= (1/ √2 ​ ) /​ ​ (√3/2) ​ ​ = √2 / √3 

Example 7: Finding tan ⁡  tanθ from sinθ= 1 ​ and cosθ= √3/2. ​ ​ 

Solution: Given: sinθ= 1

cosθ= √3/2. ​  

Using the relationship 

tanθ= sinθ / cosθ​ : 

tanθ= (1 ) /​ ​ (√3/2) ​ ​ = 2 / √3 

Example 8: Determine tanx from secx= 1/3 ​

Solution:

Given: secx= 1 /3 ​  ​

Using the identity sec ² x−tan ² x=1 and

tan ² x=sec ² x−1:

tan ² x=( 1/3  ​ ) 2 −1= 1/9 ​ −1 = (1−9) ​/ 9  =− 8/9 ​

tan(x) = √(-8/9)

tan(x) = √(-8/9)

The tangent function, an important aspect of trigonometry, serves to establish the relationship between angles and side lengths within triangles, particularly in right-angled triangles. It aids in determining unknown angles or side lengths using specific trigonometric functions and formulas. The fundamental formula for tan θ involves the ratio of the side opposite an angle to the adjacent side.

Throughout various problem-solving scenarios, the tangent function has been applied to find unknown angles or ratios. From identifying angles using known side lengths to solving for tangent values when given information about other trigonometric functions like sine, cosine, and secant, the application of the tangent function is crucial in trigonometric problem-solving.

Understanding the fundamental principles and relationships between trigonometric functions like tangent, sine, cosine, cotangent, secant, and cosecant, as well as their respective formulas, enables precise calculations and analysis within trigonometry, particularly in the context of right-angled triangles.

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