The sides of atriangle are in Arithmetic Progression (A P ) If the smallest angle of the triangle is - Maths - Solution of Triangles

Question

The sides of atriangle are in Arithmetic Progression (A P ) If the
smallest angle of the triangle is α and largest angle of the
triangle exceeds smallest angle by β , then what is value of
tan[α + (β/2)] ?

Mayank Jha · Accepted Answer

As  no other information variable is given, hence we can assume sides oftriangle
Let sides of triangle be 3,4,5
Smallest angle is α, largest angle is α+β, hence third angle γ is given byγ=π-α+α+β=π-2α+βAndα<π-2α+β<α+βSide opposite to this angle will be 4
Apply cosine rulecosπ-2α+β=32+52-422×3×5⇒-cos2α+β=9+25-1630=1830=35⇒cos2α+β=-35We know thatcos2θ=2cos2θ-1⇒cos2α+β=2cos22α+β2-1⇒cos2α+β=2cos2α+β2-1⇒-35=2cos2α+β2-1⇒2cos2α+β2=1-35=25⇒cos2α+β2=210⇒cos2α+β2=15⇒sec2α+β2=5⇒1+tan2α+β2=5⇒tan2α+β2=5-1=4⇒tanα+β2=2Note: If answer is something else then do tell me if terms of what, value oftanα+β2 is given

The sides of atriangle are in Arithmetic Progression (A.P.). If the smallest angle of the triangle is α and largest angle of the triangle exceeds smallest angle by β , then what is value of tan[α + (β/2)] ?