Question

If alpha beta are acute and are roots of 2tan^(2)theta+35tan theta+2=0 then show that alpha+beta=(pi)/(2).​

Answer 1

Step-by-step explanation:

Given :-

α, β are acute and roots of 2Tan²θ+35Tanθ+2=0.

To find :-

Show that α+β = π/2 .

Solution :-

Given Quadratic equation is 2Tan²θ+35Tanθ+2=0.

This is the equation in Tan θ

Let the roots be Tan α and Tan β

On Comparing this with the standard quadratic equation ax²+bx+c = 0

a = 2

b = 35

c = 2

We know that

Sum of the roots = -b/a

=> Tan α + Tan β = -35/2 ------------(1)

and

Product of the roots = c/a

Tan α . Tan β =2/2

=> Tan α . Tan β = 1 ----------------(2)

We know that

Tan (A+B) = (Tan A+Tan B) /(1-Tan A Tan B )

Now,

Tan (α+β) = (Tan α+ Tan β)/(1- Tan α .Tan β )

=> Tan (α+β) = (-35/2)/(1-1)

=> Tan (α+β) = (-35/2)/0

=> Tan (α+β) = not defined

=> Tan (α+β) = Tan 90°

=> α+β = 90°

=> α+β = 180°/2

=> α+β = π/2

Hence, Proved.

Answer:-

If α, β are acute and roots of 2Tan²θ+35Tanθ+2=0. then α+β = π/2

Used formulae:-

→The standard quadratic equation is ax²+bx+c = 0

→ Sum of the roots = -b/a

→ Product of the roots = c/a

→ Tan(A+B) =(Tan A+Tan B)/(1-Tan A Tan B)

→ Tan 90° = not defined

→ Division with zero is not defined.

→ π = 180°