Have a query? Ask a Tutor!!

Access to best educators and exclusive paper discussions

Offsite Campus Turito Championship

Can’t find what you’re looking for?

Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

Homework- One to One Solutions
Understand concepts to solve better
Get your doubts solved instantly

Maths-

General

Easy

Question

In $capital delta A B C comma c o t invisible function application A plus c o t invisible function application B plus c o t invisible function application C equals$

$(a + b + c) / 4 Δ$
$(a^{2} + b^{2} + c^{2}) / 4 Δ$
$((a + b + c)^{2}) / 4 Δ$
$((a + b + c)^{2}) / 2 Δ$

hint Hint:

In this question we have to use the cosine rule and sine rule.
First write all the equations of cosine rule then add them and replace the product of the sides from given formulas. $a squared equals space b squared plus c squared minus 2 b c cos open parentheses A close parentheses b squared equals a squared plus c squared minus 2 a c cos open parentheses B close parentheses c squared equals a squared plus b squared minus 2 a b cos open parentheses C close parentheses$ , $1 half a b sin open parentheses C close parentheses equals increment 1 half a c sin open parentheses B close parentheses equals increment 1 half b c sin open parentheses A close parentheses equals increment$

The correct answer is: $left parenthesis a squared plus b squared plus c squared right parenthesis divided by 4 capital delta$

In this question we have to find the value of $cot open parentheses A close parentheses plus cot open parentheses B close parentheses plus cot open parentheses C close parentheses$
We know that in a triangle
$1 half a b sin open parentheses C close parentheses equals increment 1 half a c sin open parentheses B close parentheses equals increment 1 half b c sin open parentheses A close parentheses equals increment$
By Using Cosine Rule
$a squared equals space b squared plus c squared minus 2 b c cos open parentheses A close parentheses b squared equals a squared plus c squared minus 2 a c cos open parentheses B close parentheses c squared equals a squared plus b squared minus 2 a b cos open parentheses C close parentheses$
By adding above three equations we get,
$a squared plus b squared plus c squared equals 2 b c cos open parentheses A close parentheses plus 2 a c cos open parentheses B close parentheses plus 2 a b cos open parentheses C close parentheses$
We can replace $b c comma a c comma a b$ by $fraction numerator 2 increment over denominator sin open parentheses A close parentheses end fraction comma fraction numerator 2 increment over denominator sin open parentheses B close parentheses end fraction comma fraction numerator 2 increment over denominator sin open parentheses C close parentheses end fraction$
By substituting we get,
$a squared plus b squared plus c squared equals fraction numerator 4 increment cos open parentheses A close parentheses over denominator sin open parentheses A close parentheses end fraction plus fraction numerator 4 increment cos open parentheses B close parentheses over denominator sin open parentheses B close parentheses end fraction plus fraction numerator 4 increment cos open parentheses C close parentheses over denominator sin open parentheses C close parentheses end fraction$
We know that $cot open parentheses x close parentheses equals fraction numerator cos open parentheses x close parentheses over denominator sin open parentheses x close parentheses end fraction$
=> $fraction numerator a squared plus b squared plus c squared over denominator 4 increment end fraction equals cot open parentheses A close parentheses plus cot open parentheses B close parentheses plus cot open parentheses C close parentheses$

Related Questions to study

In $triangle A B C comma 4 R capital delta sin invisible function application A divided by 2 space sin invisible function application B divided by 2 space sin invisible function application C divided by 2 equals$

In $triangle A B C comma 4 R capital delta sin invisible function application A divided by 2 space sin invisible function application B divided by 2 space sin invisible function application C divided by 2 equals$

If 'O' is the Orthocentre of $triangle A B C$ then $O A equals$

If 'O' is the Orthocentre of $triangle A B C$ then $O A equals$

In $I n triangle A B C comma sum a cubed c o s invisible function application left parenthesis B minus C right parenthesis equals$

In $I n triangle A B C comma sum a cubed c o s invisible function application left parenthesis B minus C right parenthesis equals$

parallel

In $triangle A B C comma left parenthesis a plus b right parenthesis cos invisible function application C plus left parenthesis b plus c right parenthesis cos invisible function application A plus left parenthesis c plus a right parenthesis cos invisible function application B equals$

In $triangle A B C comma left parenthesis a plus b right parenthesis cos invisible function application C plus left parenthesis b plus c right parenthesis cos invisible function application A plus left parenthesis c plus a right parenthesis cos invisible function application B equals$

In $triangle A B C comma sum 1 divided by a c o s squared invisible function application A divided by 2 equals$

In $triangle A B C comma sum 1 divided by a c o s squared invisible function application A divided by 2 equals$

In $triangle A B C comma s squared t a n invisible function application A divided by 2 t a n invisible function application B divided by 2 t a n invisible function application C divided by 2 equals$

In $triangle A B C comma s squared t a n invisible function application A divided by 2 t a n invisible function application B divided by 2 t a n invisible function application C divided by 2 equals$

parallel

In $triangle A B C comma 4 R s sin invisible function application A divided by 2 space sin invisible function application B divided by 2 space sin invisible function application C divided by 2 equals$

In $triangle A B C comma 4 R s sin invisible function application A divided by 2 space sin invisible function application B divided by 2 space sin invisible function application C divided by 2 equals$

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

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

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

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

parallel

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

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

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

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

parallel

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 $f left parenthesis x right parenthesis equals x t a n left parenthesis negative 1 right parenthesis invisible function application x$ 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 $L t left parenthesis x rightwards arrow infinity right parenthesis space space left parenthesis f left parenthesis 2 x right parenthesis right parenthesis divided by left parenthesis f left parenthesis x right parenthesis right parenthesis$

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

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

parallel

card img

With Turito Academy.

card img

With Turito Foundation.

card img

Get an Expert Advice From Turito.

Turito Academy

card img

With Turito Academy.

Test Prep

card img

With Turito Foundation.