It is quadratic, no it's trigonometry

Geometry Level 3

Let △ A B C \triangle ABC be a triangle right angled at C C . If tan ⁡ A 2 \tan\frac { A }{ 2 } and tan ⁡ B 2 \tan\frac { B }{ 2 } are the roots of the quadratic equation a x 2 + b x + c = 0 ax ^{ 2 }+bx+c=0 , then which of the following is true?

a+b+c=0 b+c=a a-b=c a+b=c

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Gautam Jha by Gautam Jha , 21, India

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1 solution

Marta Reece
Jun 12, 2017

B 2 = 4 5 ∘ − A 2 \dfrac B2=45^\circ-\dfrac A2

tan ⁡ B 2 = tan ⁡ ( 4 5 ∘ − A 2 ) = 1 − tan ⁡ A 2 1 + tan ⁡ A 2 \tan \frac B2=\tan(45^\circ-\frac A2)=\dfrac{1-\tan \frac A2}{1+\tan\frac A2}

If they are roots of a quadratic equation, they can be expressed in terms of the coefficients and the result substituted into this equation

− b + b 2 − 4 c 2 a = 1 − − b − b 2 − 4 c 2 a 1 + − b − b 2 − 4 c 2 a \dfrac{-b+\sqrt{b^2-4c}}{2a}=\dfrac{1-\frac{-b-\sqrt{b^2-4c}}{2a}}{1+\frac{-b-\sqrt{b^2-4c}}{2a}}

This simplifies to c = a + b c=a+b

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