Answer each of the following questions in one word or one sentence or as per exact requirement of for question-In a - ABC- if sin A and sin B are the roots of the equation c 2 x 2- c a - b x - ab -0- then find - C-

C^2x^2 c(a+b )x+ab=0 sin A and sin B are the roots of the equation, sin A+sin B=c(a+b )/c^2=(a+b )/c sin A sin B=ab/c^2 sin A sin B =√((sin A+sin B )^2 ...

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Byju's Answer

Standard XII

Mathematics

Solution of Triangle

Answer each o...

Question

Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$I n a Δ A B C, i f s i n A a n d s i n B are the roots of the equation c^{2} x^{2} - c (a + b) x + a b = 0, t h e n f i n d ∠ C .$

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Solution

$c^{2} x^{2} - c (a + b) x + a b = 0 sin A and sin B are the roots of the equation, s i n A + s i n B = \frac{c (a + b)}{c^{2}} = \frac{(a + b)}{c} s i n A s i n B = \frac{a b}{c^{2}} s i n A - s i n B = \sqrt{{(s i n A + s i n B)}^{2} - 4 s i n A . s i n B} = \sqrt{{(\frac{a + b}{c})}^{2} - 4 \frac{a b}{c^{2}}} = \sqrt{\frac{a^{2} + b^{2} + 2 a b - 4 a b}{c^{2}}} = \sqrt{\frac{a^{2} + b^{2} - 2 a b}{c^{2}}} = \sqrt{\frac{{(a - b)}^{2}}{c^{2}}} = \frac{l e f t (a - b)}{c} s i n A + s i n B = \frac{(a + b)}{c} . . . (i) s i n A - s i n B = \frac{(a - b)}{c} . . . (i i) Adding (i) and (ii), we get, s i n A = \frac{a}{c} a n d s i n B = \frac{b}{c} s i n A = \frac{P e r p e n d i c u l a r}{h y p o t e n u s e} S o, I n Δ A B C, one of the side = a, another side = b, Third side = c ∴ a^{2} + b^{2} = c^{2} ∠ C = 90^{\circ}$

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