Three positive acute angles α, β and γ satisfy the relation tan (β)/(2) = (1)/(3)cot (α)/(2) and cot (γ)/(2) = (1)/(2) (3tan (α)/(2) + cot (α)/(2)). Then, the value of α + β + γ is equal to

Correct answer: A. π We have,& tan (α)/(2)tan (β)/(2) = (1)/(3) ⇒ 1 - tan (α)/(2)tan (β)/(2) = (2)/(3) Now, ((α + β)/(2)) = (tan (α)/(2) + tan (β)/(2))/(1 - tan (α)/(2)tan (β)/(2)) = (tan (α)/(2) + tan (β)/(2))/((2)/(3)) ⇒ ((α + β)/(2)) = (3)/(2) ( tan (α)/(2) + tan (β)/(2)) =(3)/(2) tan (α)/(2) + (1)/(2)(3tan (β)/(2)) = (3)/(2) tan (α)/(2) + (1)/(2) cot (α)/(2) = cot (γ)/(2) ⇒ ((α + β)/(2)) = cot (γ)/(2) = (1)/(tan (γ)/(2)) ⇒ 1 - ((α + β)/(2)) ⋅ tan (γ)/(2) =0 ⇒ ((α + β)/(2)) + (γ)/(2) = (π)/(2) ⇒ α + β + γ = π

NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

NTA Abhyas 2020 · Mathematics · Trigonometric Ratios & Identities

Three positive acute angles α, β and γ satisfy the relation tan...

Three positive acute angles

\alpha, \beta

and

\gamma

satisfy the relation

tan \dfrac{\beta}{2} = \dfrac{1}{3}cot \dfrac{\alpha}{2}

and

cot \dfrac{\gamma}{2} = \dfrac{1}{2} \left(3tan \dfrac{\alpha}{2} +

cot

\dfrac{\alpha}{2}\right). Then, the value of

\alpha + \beta + \gamma is

equal to

Options

A $\pi$
B $2\pi$
C $\dfrac{\pi}{2}$
D $\dfrac{3\pi}{2}$

Correct answer

A. $\pi$

Step-by-step solution

We have,&

tan \dfrac{\alpha}{2}tan \dfrac{\beta}{2} = \dfrac{1}{3}

⇒ 1 - tan \dfrac{\alpha}{2}tan \dfrac{\beta}{2} = \dfrac{2}{3}

Now,

\tan \left(\dfrac{\alpha + \beta}{2}\right) = \dfrac{tan \dfrac{\alpha}{2} + tan \dfrac{\beta}{2}}{1 - tan \dfrac{\alpha}{2}tan \dfrac{\beta}{2}} = \dfrac{tan \dfrac{\alpha}{2} + tan \dfrac{\beta}{2}}{\dfrac{2}{3}}

⇒ \tan \left(\dfrac{\alpha + \beta}{2}\right) = \dfrac{3}{2} \left(

tan

\dfrac{\alpha}{2} +

tan

\dfrac{\beta}{2}\right)<br />=\dfrac{3}{2}

tan

\dfrac{\alpha}{2} + \dfrac{1}{2}\left(3tan \dfrac{\beta}{2}\right) = \dfrac{3}{2}

tan

\dfrac{\alpha}{2} + \dfrac{1}{2}

cot

\dfrac{\alpha}{2}<br />

= cot

\dfrac{\gamma}{2}<br />

⇒ \tan \left(\dfrac{\alpha + \beta}{2}\right) =

cot

\dfrac{\gamma}{2} = \dfrac{1}{tan \dfrac{\gamma}{2}}

⇒ 1 - \tan \left(\dfrac{\alpha + \beta}{2}\right)

⋅ tan

\dfrac{\gamma}{2} =0

⇒ \left(\dfrac{\alpha + \beta}{2}\right) + \dfrac{\gamma}{2} = \dfrac{\pi}{2}

⇒ \alpha + \beta + \gamma = \pi$$

Practice Trigonometric Ratios & Identities on Quantrex Academy →

More from Trigonometric Ratios & Identities

Let α = (1 - 2 ((π)/(11)))(1 - 2 ((3π)/(11)))(1 - 2 ((9π)/(11)))(1 - 2 ((27π)/(11)))(1… JEE Advanced 2026 If A = ( 3^∘)/( 9^∘) + ( 9^∘)/( 27^∘) + ( 27^∘)/( 81^∘) and B = 81^∘ - 3^∘ , then (B)/(A)… JEE Main 2026 Let P = \ θ ∈ [0, 4π] : ^2θ ≠ 1\ and S = \ a ∈ ℤ : 2( ^8θ - ^8θ) 2θ = a^2, θ ∈ P\ . Then… JEE Main 2026 If ((π)/(18)) ((5π)/(18)) ((7π)/(18)) = K , then the value of ((10Kπ)/(3)) is : JEE Main 2026 If (A-B) A + ^ 2 C ^ 2 ~A =1, ~A, ~B, C ∈(0, (π)/(2)) , then JEE Main 2026 If x=(5)/(12) for some x ∈(π, (3 π)/(2)) , then 7 x( (13 x)/(2)+ (13 x)/(2))+ 7 x( (13… JEE Main 2026 The value of 3 cosec 20^ ∘ - 20^ ∘ 20^ ∘ 40^ ∘ 60^ ∘ 80^ ∘ is equal to JEE Main 2026 The least value of ( ^ 2 θ-6 θ θ+3 ^ 2 θ+2) is JEE Main 2026 Let (π)/(2) ((15 θ)/(2))( 8 θ+ 8 θ)+ ((15 θ)/(2))( 8 θ- 8 θ) is equal to JEE Main 2026 Let α and β respectively be the maximum and the minimum values of the function f(θ)=4( ^… JEE Main 2026 Let (α+β)=-(1)/(10) and (α-β)=(3)/(8) , where 0<α<(π)/(3) and 0<β<(π)/(4) . If 2 α= 3(1-r… JEE Main 2026 If ^ 2 48^ ∘ - ^ 2 12^ ∘ ^ 2 24^ ∘ - ^ 2 6^ ∘ = α+β 5 2 , where α, β ∈ ℕ , then α+β is… JEE Main 2026 The value of cosec 10^ ∘ - 3 10^ ∘ is equal to : JEE Main 2026 Let α= 1 60^ ∘ 61^ ∘ + 1 62^ ∘ 63^ ∘ +…+ 1 118^ ∘ 119^ ∘ Then the value of ( cosec 1^ ∘ α… JEE Advanced 2025 If x+ ^2 x=1, x ∈(0, (π)/(2)) , then ( ^ 12 x+ ^ 12 x)+3( ^ 10 x+ ^ 10 x+ ^8 x+ ^8 x)+(… JEE Main 2025 If ∑_ r=1 ^ 13 \ (1)/( ((π)/(4)+(r-1) (π)/(6)) ((π)/(4)+(r π)/(6)))\ =a 3 +b, a, b ∈ Z… JEE Main 2025 Full Trigonometric Ratios & Identities list Trigonometric Ratios & Identities JEE Maths PYQs JEE Mathematics PYQs All NTA Abhyas JEE Main PYQs