If in a - ABC - 2 b 2- a 2- c 2- then sin 3 B -sin B is equal toA- c2-a2-22-a2B- c 2- a 2-2 caC- c2-a2-c a2D- c2-a2-2 c a2

The correct option is D (c^2 a^2/2ca )^2 sin 3B/sin B=3sinB 4sin^3B/sinB =3 4sin^2B = 1+4(a^2+c^2 b^2/2ac)^2 = 1+4(a^2+c^2 b^2)^2/4(ac)^2 = 1+a^2+c^2 a^2+c ...

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Byju's Answer

Standard XII

Mathematics

Cosine Rule

If in a Δ ABC...

Question

If in a $Δ$ ABC, $2 b^{2} = a^{2} + c^{2}$ , then $\frac{s i n 3 B}{s i n B}$ is equal to

$\frac{c^{2} - a^{2}}{2 c a}$

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$\frac{c^{2} - a^{2}}{2 c a}$

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${(\frac{c^{2} - a^{2}}{c a})}^{2}$

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${(\frac{c^{2} - a^{2}}{2 c a})}^{2}$

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Solution

The correct option is D
${(\frac{c^{2} - a^{2}}{2 c a})}^{2}$

$\frac{s i n 3 B}{s i n B} = \frac{3 s i n B - 4 s i n^{3} B}{s i n B} = 3 - 4 s i n^{2} B = - 1 + 4 {(\frac{a^{2} + c^{2} - b^{2}}{2 a c})}^{2} = - 1 + 4 \frac{(a^{2} + c^{2} - b^{2})^{2}}{4 (a c)^{2}} = - 1 + \frac{a^{2} + c^{2} - \frac{a^{2} + c^{2}}{2}}{(a c)^{2}} (g i v e n 2 b^{2} = a^{2} + c^{2}) = - 1 + \frac{{(\frac{a^{2} + c^{2}}{2})}^{2}}{(a c)^{2}} = \frac{- 4 a^{2} c^{2} + a^{4} + c^{4} + 2 a^{2} c^{2}}{4 a^{2} c^{2}} = \frac{(c^{2} - a^{2})^{2}}{4 (a c)^{2}} = {(\frac{c^{2} - a^{2}}{2 a c})}^{2}$

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Similar questions

Q. If in

Δ A B C,

2 b^{2} = a^{2} + c^{2}

, then

\frac{sin 3 B}{sin B}

is equal to

Q. If

a^{2} + c^{2} = 2 b^{2}

where a, b, c are the sides of a triangle, then prove that

\frac{s i n 3 B}{s i n B} = [\frac{a^{2} - c^{2}}{2 a c}]^{2}

Q. In an acute angle triangle

A B C,

\frac{sin 3 B}{sin B} = {(\frac{a^{2} - c^{2}}{2 a c})}^{2},

then

a^{2}, b^{2}, c^{2}

are in

If \frac{S i n 3 B}{S i n B} = (\frac{a^{2} - c^{2}}{2 a c})^{2}

then

a^{2}, b^{2}, c^{2}

are in:

In a $△ A B C, 2 c a s i n (\frac{A - B + C}{2})$ is equal to

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If in a Δ ABC, 2b2=a2+c2, then sin3BsinB is equal to

The correct option is D (c2−a22ca)2 sin3BsinB=3sinB−4sin3BsinB=3−4sin2B=−1+4(a2+c2−b22ac)2=−1+4(a2+c2−b2)24(ac)2=−1+a2+c2−a2+c22(ac)2 (given 2b2=a2+c2)=−1+(a2+c22)2(ac)2=−4a2c2+a4+c4+2a2c24a2c2=(c2−a2)24(ac)2=(c2−a22ac)2

If in a $Δ$ ABC, $2 b^{2} = a^{2} + c^{2}$ , then $\frac{s i n 3 B}{s i n B}$ is equal to