Question

Q. If α, β and γ are positive active angles, such that
$\sin( \alpha + \beta - \gamma ) = \frac{1}{2}$,
$\cos( \beta + \gamma - \alpha ) = \frac{1}{2}$and
$\tan( \gamma + \alpha - \beta ) = 1$, then find the values of α, β and γ .

##NO USELESS ANS. PLZ.##

Answer 1

hey...

here is ur solution

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#APS☺

Answer 2

♧♧HERE IS YOUR ANSWER♧♧

Trigonometry is the study of angles and its ratios. This study prescribes the relation amongst the sides of a triangle and angles of the triangle.

There are sine, cosine, tangent, cot, sectant and cosec ratios to an angle of a triangle.

Let me tell you an interesting fact about Trigonometry.

"Triangle" > "Trigonometry"

Remember some formulae now :

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1

Want to learn more!

Here it is :

sin(A + B) = sinA cosB + cosA sinB

sin(A - B) = sinA cosB - cosA sinB

cos(A + B) = cosA cosB - sinA sinB

cos(A - B) = cosA cosB + sinA sinB

SOLUTION :

Given,

sin (α + β - γ) = ½
=> sin (α + β - γ) = sin 30
=> α + β - γ = 30 .....(i)

cos (β + γ - α) = ½
=> cos (β + γ - α) = cos 60
=> β + γ - α = 60 .....(ii)

and

tan (γ + α - β) = 1
=> tan (γ + α - β) = tan 45
=> γ + α - β = 45 .....(iii)

Now, adding (i), (ii) and (iii), we get :

α + β + γ = 135 .....(iv)

Now, (i) - (iv) gives

- 2γ = - 105
=> γ = 105/2

Also, (ii) - (iv) gives

- 2α = - 75
=> α = 75/2

Putting α = 75/2 and γ = 105/2 in (iii), we get :

105/2 + 75/2 - β = 45

=> β = 45

Therefore the values of α, β and γ are 75/2°, 45° and 105/2° respectively.

Thank you for your question.