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Question

In $A B C space comma space 4 R cos A divided by 2 space cos B divided by 2 space cos C divided by 2 equals$

Δ D
S
r
2s

hint Hint:

In this question we have to use the formula of solution of a triangle.
$cos open parentheses A over 2 close parentheses equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator b c end fraction end root comma space cos open parentheses B over 2 close parentheses equals square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator a c end fraction end root comma cos open parentheses C over 2 close parentheses equals space square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator a b end fraction end root$ and $S space equals square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root$ and $fraction numerator a b c over denominator 4 R end fraction equals space S$

The correct answer is: S

In this question we have to find the value of $4 R cos open parentheses A over 2 close parentheses cos open parentheses B over 2 close parentheses cos open parentheses C over 2 close parentheses$
We know that $cos open parentheses A over 2 close parentheses equals square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator b c end fraction end root comma space cos open parentheses B over 2 close parentheses equals square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator a c end fraction end root comma cos open parentheses C over 2 close parentheses equals space square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator a b end fraction end root$
=> $4 R cos open parentheses A over 2 close parentheses cos open parentheses B over 2 close parentheses cos open parentheses C over 2 close parentheses$
=> $4 R square root of fraction numerator s left parenthesis s minus a right parenthesis over denominator b c end fraction end root cross times square root of fraction numerator s left parenthesis s minus b right parenthesis over denominator a c end fraction end root cross times square root of fraction numerator s left parenthesis s minus c right parenthesis over denominator a b end fraction end root$
=> $4 R square root of fraction numerator s cubed left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis over denominator a squared b squared c squared end fraction end root$
=> $fraction numerator 4 R s over denominator a b c end fraction square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root$
We know that $S space equals square root of s left parenthesis s minus a right parenthesis left parenthesis s minus b right parenthesis left parenthesis s minus c right parenthesis end root$ and $fraction numerator a b c over denominator 4 R end fraction equals space S$
=> $1 over S cross times s cross times S$
=> $s$

Related Questions to study

In $triangle A B C comma sum a cos invisible function application A equals$

In $triangle A B C comma sum a cos invisible function application A equals$

In $capital delta A B C comma sum left parenthesis left parenthesis b squared minus c squared right parenthesis right parenthesis divided by a c o s invisible function application A equals$

In $capital delta A B C comma sum left parenthesis left parenthesis b squared minus c squared right parenthesis right parenthesis divided by a c o s invisible function application A equals$

$L t left parenthesis x rightwards arrow pi divided by 6 right parenthesis left parenthesis left parenthesis 3 s i n x minus √ 3 c o s x right parenthesis divided by left parenthesis 6 x minus pi right parenthesis right parenthesis equals$

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parallel

If $L t left parenthesis x rightwards arrow 0 right parenthesis left parenthesis left parenthesis c o s invisible function application 4 x plus a c o s invisible function application 2 x plus b right parenthesis divided by x to the power of 4 right parenthesis$ is finite then the values of a,b are respectively

If $L t left parenthesis x rightwards arrow 0 right parenthesis left parenthesis left parenthesis c o s invisible function application 4 x plus a c o s invisible function application 2 x plus b right parenthesis divided by x to the power of 4 right parenthesis$ is finite then the values of a,b are respectively

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If $f left parenthesis x right parenthesis equals x t a n left parenthesis negative 1 right parenthesis invisible function application x$ then $l i m subscript left parenthesis x rightwards arrow 1 right parenthesis left parenthesis f left parenthesis x right parenthesis minus f left parenthesis 1 right parenthesis right parenthesis divided by left parenthesis x minus 1 right parenthesis equals$

If $f colon R rightwards arrow R$ be a positive increasing function with $L t left parenthesis x rightwards arrow infinity right parenthesis space space left parenthesis f left parenthesis 3 x right parenthesis right parenthesis divided by left parenthesis f left parenthesis x right parenthesis right parenthesis equals 1 space$ then $L t left parenthesis x rightwards arrow infinity right parenthesis space space left parenthesis f left parenthesis 2 x right parenthesis right parenthesis divided by left parenthesis f left parenthesis x right parenthesis right parenthesis$

If $f colon R rightwards arrow R$ be a positive increasing function with $L t left parenthesis x rightwards arrow infinity right parenthesis space space left parenthesis f left parenthesis 3 x right parenthesis right parenthesis divided by left parenthesis f left parenthesis x right parenthesis right parenthesis equals 1 space$ then $L t left parenthesis x rightwards arrow infinity right parenthesis space space left parenthesis f left parenthesis 2 x right parenthesis right parenthesis divided by left parenthesis f left parenthesis x right parenthesis right parenthesis$

parallel

$L t left parenthesis n rightwards arrow infinity right parenthesis left square bracket 1 divided by n squared s e c squared 1 divided by n squared plus 2 divided by n squared s e c squared 4 divided by n squared plus midline horizontal ellipsis. plus 1 divided by n s e c squared 1 right square bracket equals$

$L t left parenthesis n rightwards arrow infinity right parenthesis left square bracket 1 divided by n squared s e c squared 1 divided by n squared plus 2 divided by n squared s e c squared 4 divided by n squared plus midline horizontal ellipsis. plus 1 divided by n s e c squared 1 right square bracket equals$

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parallel

$L t ┬ left parenthesis n rightwards arrow infinity right parenthesis left parenthesis 1 to the power of p plus 2 to the power of p plus 3 to the power of p plus midline horizontal ellipsis. plus n to the power of p right parenthesis divided by n to the power of left parenthesis p plus 1 right parenthesis$ is

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For $x element of R L t left parenthesis x rightwards arrow infinity right parenthesis left parenthesis left parenthesis x minus 3 right parenthesis divided by left parenthesis x plus 2 right parenthesis right parenthesis to the power of x equals$

For $x element of R L t left parenthesis x rightwards arrow infinity right parenthesis left parenthesis left parenthesis x minus 3 right parenthesis divided by left parenthesis x plus 2 right parenthesis right parenthesis to the power of x equals$

$L t left parenthesis h rightwards arrow 0 right parenthesis left parenthesis f left parenthesis 2 h plus 2 plus h squared right parenthesis minus f left parenthesis 2 right parenthesis right parenthesis divided by left parenthesis f left parenthesis h minus h squared plus 1 right parenthesis minus f left parenthesis 1 right parenthesis right parenthesis$ , given that $f to the power of apostrophe left parenthesis 2 right parenthesis equals 6$ and $f to the power of apostrophe left parenthesis 1 right parenthesis equals 4$

$L t left parenthesis h rightwards arrow 0 right parenthesis left parenthesis f left parenthesis 2 h plus 2 plus h squared right parenthesis minus f left parenthesis 2 right parenthesis right parenthesis divided by left parenthesis f left parenthesis h minus h squared plus 1 right parenthesis minus f left parenthesis 1 right parenthesis right parenthesis$ , given that $f to the power of apostrophe left parenthesis 2 right parenthesis equals 6$ and $f to the power of apostrophe left parenthesis 1 right parenthesis equals 4$

parallel

$L i m left parenthesis x rightwards arrow 0 right parenthesis left parenthesis x t a n 2 x minus 2 x t a n x right parenthesis divided by left parenthesis left parenthesis 1 minus c o s invisible function application 2 x right parenthesis squared right parenthesis equals$

$L i m left parenthesis x rightwards arrow 0 right parenthesis left parenthesis x t a n 2 x minus 2 x t a n x right parenthesis divided by left parenthesis left parenthesis 1 minus c o s invisible function application 2 x right parenthesis squared right parenthesis equals$

If $f left parenthesis a right parenthesis equals 2 comma f to the power of apostrophe left parenthesis a right parenthesis equals 1 comma g left parenthesis a right parenthesis equals negative 1 comma g to the power of factorial left parenthesis a right parenthesis equals 2 comma$ then the value of $L i m subscript left parenthesis x rightwards arrow a right parenthesis invisible function application left parenthesis g left parenthesis x right parenthesis f left parenthesis a right parenthesis minus g left parenthesis a right parenthesis f left parenthesis x right parenthesis right parenthesis divided by left parenthesis x minus a right parenthesis$ is
,

If $f left parenthesis a right parenthesis equals 2 comma f to the power of apostrophe left parenthesis a right parenthesis equals 1 comma g left parenthesis a right parenthesis equals negative 1 comma g to the power of factorial left parenthesis a right parenthesis equals 2 comma$ then the value of $L i m subscript left parenthesis x rightwards arrow a right parenthesis invisible function application left parenthesis g left parenthesis x right parenthesis f left parenthesis a right parenthesis minus g left parenthesis a right parenthesis f left parenthesis x right parenthesis right parenthesis divided by left parenthesis x minus a right parenthesis$ is
,

If $G left parenthesis x right parenthesis equals negative √ left parenthesis 25 minus x squared right parenthesis$ then $L i m subscript left parenthesis x rightwards arrow 1 right parenthesis invisible function application left parenthesis G left parenthesis x right parenthesis minus G left parenthesis 1 right parenthesis right parenthesis divided by left parenthesis x minus 1 right parenthesis$ has the value

If $G left parenthesis x right parenthesis equals negative √ left parenthesis 25 minus x squared right parenthesis$ then $L i m subscript left parenthesis x rightwards arrow 1 right parenthesis invisible function application left parenthesis G left parenthesis x right parenthesis minus G left parenthesis 1 right parenthesis right parenthesis divided by left parenthesis x minus 1 right parenthesis$ has the value

parallel

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