Maths-
General
Easy

Question

In straight triangle A B C comma open parentheses fraction numerator b minus c over denominator a end fraction close parentheses sin invisible function application open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses equals

  1. cos invisible function application open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses
  2. sin invisible function application open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses
  3. sin invisible function application open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses
  4. cos invisible function application open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses

Hint:

In the given question we will first use the sine rulefraction numerator a over denominator sin space A end fraction equals fraction numerator b over denominator sin space B end fraction equals fraction numerator c over denominator sin space C end fraction. Then after putting the value of a, b and c in terms of sin we will simplify using the trigonometric identities.

The correct answer is: sin invisible function application open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses


    By the sine rule, we have
    fraction numerator a over denominator sin space A end fraction equals fraction numerator b over denominator sin space B end fraction equals fraction numerator c over denominator sin space C end fraction
    let fraction numerator a over denominator sin space A end fraction equals fraction numerator b over denominator sin space B end fraction equals fraction numerator c over denominator sin space C end fraction =k
    rightwards double arrowa = k sin A, b =k sin B and c = k sin C
    Now, open parentheses fraction numerator b minus c over denominator a end fraction close parentheses space sin space fraction numerator B plus C over denominator 2 end fraction
    equals open parentheses fraction numerator k space sin space B space minus space k space sin space C space over denominator k sin space A end fraction close parentheses sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses
equals open parentheses fraction numerator space sin space B space minus space space sin space C space over denominator sin space A end fraction close parentheses sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses
equals fraction numerator 2 cos open parentheses begin display style fraction numerator B plus C over denominator 2 end fraction end style close parentheses sin open parentheses begin display style fraction numerator B minus C over denominator 2 end fraction end style close parentheses over denominator 2 sin begin display style A over 2 end style cos begin display style A over 2 end style end fraction sin open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses space space space space space space space space space space space space space space space space space space space space space space open square brackets sin space A minus space sin space B equals 2 space cos space open parentheses fraction numerator A plus B over denominator 2 end fraction close parentheses space sin open parentheses fraction numerator A minus B over denominator 2 end fraction close parentheses close square brackets
equals fraction numerator cos open parentheses begin display style straight pi over 2 end style minus begin display style A over 2 end style close parentheses sin space open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses sin space open parentheses begin display style straight pi over 2 end style minus begin display style A over 2 end style close parentheses over denominator sin A over 2 cos A over 2 end fraction space space space space space space space space space space space space space space space space space space space space space open square brackets A plus B plus C equals straight pi comma space fraction numerator straight A plus straight B plus straight C over denominator 2 end fraction equals straight pi over 2 comma space fraction numerator straight B plus straight C over denominator 2 end fraction equals straight pi over 2 minus straight A over 2 close square brackets
equals fraction numerator sin begin display style A over 2 end style sin begin display style fraction numerator B minus C over denominator 2 end fraction end style cos begin display style A over 2 end style over denominator sin begin display style A over 2 end style cos begin display style A over 2 end style end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets cos open parentheses straight pi over 2 minus A over 2 close parentheses equals sin A over 2 comma sin space open parentheses straight pi over 2 minus A over 2 close parentheses equals cos A over 2 close square brackets
equals sin open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses

    Book A Free Demo

    +91

    Grade*

    Related Questions to study

    General
    physics-

    An antenna behaves as resonant circuit only when its length is

    An antenna behaves as resonant circuit only when its length is

    physics-General
    General
    Maths-

    If A equals 60 to the power of ring operator text  then the value of  end text open parentheses 1 plus a over c plus b over c close parentheses open parentheses 1 plus c over b minus a over b close parentheses equals

    If A equals 60 to the power of ring operator text  then the value of  end text open parentheses 1 plus a over c plus b over c close parentheses open parentheses 1 plus c over b minus a over b close parentheses equals
    open parentheses fraction numerator c plus a plus b over denominator c end fraction close parentheses open parentheses fraction numerator b plus c minus a over denominator b end fraction close parentheses
equals fraction numerator b c plus c squared minus a c plus a b plus a c minus a squared plus b squared plus b c minus a b over denominator c b end fraction
equals fraction numerator 2 b c plus c squared plus b squared minus a squared over denominator c b end fraction
equals fraction numerator 2 b c plus 2 b c space cos space A over denominator b c end fraction space space left parenthesis b y space cos i n e space f o r m u l a comma space a squared equals b squared plus c squared minus 2 b c space cos A right parenthesis
equals 2 plus 2 space cos A
p u t t i n g space A equals 60 degree
equals 2 plus 2 space cos space 60 degree
equals 2 plus 2 cross times 1 half
equals 2 plus 1
equals 3

    If A equals 60 to the power of ring operator text  then the value of  end text open parentheses 1 plus a over c plus b over c close parentheses open parentheses 1 plus c over b minus a over b close parentheses equals

    Maths-General
    If A equals 60 to the power of ring operator text  then the value of  end text open parentheses 1 plus a over c plus b over c close parentheses open parentheses 1 plus c over b minus a over b close parentheses equals
    open parentheses fraction numerator c plus a plus b over denominator c end fraction close parentheses open parentheses fraction numerator b plus c minus a over denominator b end fraction close parentheses
equals fraction numerator b c plus c squared minus a c plus a b plus a c minus a squared plus b squared plus b c minus a b over denominator c b end fraction
equals fraction numerator 2 b c plus c squared plus b squared minus a squared over denominator c b end fraction
equals fraction numerator 2 b c plus 2 b c space cos space A over denominator b c end fraction space space left parenthesis b y space cos i n e space f o r m u l a comma space a squared equals b squared plus c squared minus 2 b c space cos A right parenthesis
equals 2 plus 2 space cos A
p u t t i n g space A equals 60 degree
equals 2 plus 2 space cos space 60 degree
equals 2 plus 2 cross times 1 half
equals 2 plus 1
equals 3
    General
    physics-

    In the adjoining diagram, a wavefront AB, moving in air is incident on a plane glass surface XY. Its position CD after refraction through a glass slab is shown also along with the normals drawn at A and D. The refractive index of glass with respect to air (mu equals 1) will be equal to

    In the case of refraction if CD is the refracted wave front and v1 and v2 are the speed of light in the two media, then in the time the wavelets from B reaches C, the wavelet from A will reach D, such that

    t equals fraction numerator B C over denominator v subscript a end subscript end fraction equals fraction numerator A D over denominator v subscript g end subscript end fraction rightwards double arrow fraction numerator B C over denominator A D end fraction equals fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction .....(i)
    But in capital delta A C B comma B C equals A C sin invisible function application theta .....(ii)
    while in capital delta A C D comma A D equals A C sin invisible function application phi to the power of ´ end exponent .....(iii)
    From equations (i), (ii) and (iii) fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction
    Also mu proportional to fraction numerator 1 over denominator v end fraction rightwards double arrow fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction equals fraction numerator mu subscript g end subscript over denominator mu subscript a end subscript end fraction equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction rightwards double arrow mu subscript g end subscript equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction

    In the adjoining diagram, a wavefront AB, moving in air is incident on a plane glass surface XY. Its position CD after refraction through a glass slab is shown also along with the normals drawn at A and D. The refractive index of glass with respect to air (mu equals 1) will be equal to

    physics-General
    In the case of refraction if CD is the refracted wave front and v1 and v2 are the speed of light in the two media, then in the time the wavelets from B reaches C, the wavelet from A will reach D, such that

    t equals fraction numerator B C over denominator v subscript a end subscript end fraction equals fraction numerator A D over denominator v subscript g end subscript end fraction rightwards double arrow fraction numerator B C over denominator A D end fraction equals fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction .....(i)
    But in capital delta A C B comma B C equals A C sin invisible function application theta .....(ii)
    while in capital delta A C D comma A D equals A C sin invisible function application phi to the power of ´ end exponent .....(iii)
    From equations (i), (ii) and (iii) fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction
    Also mu proportional to fraction numerator 1 over denominator v end fraction rightwards double arrow fraction numerator v subscript a end subscript over denominator v subscript g end subscript end fraction equals fraction numerator mu subscript g end subscript over denominator mu subscript a end subscript end fraction equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction rightwards double arrow mu subscript g end subscript equals fraction numerator sin invisible function application theta over denominator sin invisible function application phi to the power of ´ end exponent end fraction
    General
    chemistry-

    Whicstatemeninocorrect for n=5,m=2:

    Whicstatemeninocorrect for n=5,m=2:

    chemistry-General
    General
    chemistry-

    In P-atom find out the no. of paired electrons for l=1 and m =0-

    In P-atom find out the no. of paired electrons for l=1 and m =0-

    chemistry-General
    General
    chemistry-

    For H- atomthenergrequired fothremoval electrofrovariousub-shelligiveaunder:
    The order of the energies would be:

    For H- atomthenergrequired fothremoval electrofrovariousub-shelligiveaunder:
    The order of the energies would be:

    chemistry-General
    General
    physics-

    Immiscible transparent liquids A, B, C, D and E are placed in a rectangular container of glass with the liquids making layers according to their densities. The refractive index of the liquids are shown in the adjoining diagram. The container is illuminated from the side and a small piece of glass having refractive index 1.61 is gently dropped into the liquid layer. The glass piece as it descends downwards will not be visible in

    Refractive index of liquid C is same as that of glass piece. So it will not be visible in liquid C.

    Immiscible transparent liquids A, B, C, D and E are placed in a rectangular container of glass with the liquids making layers according to their densities. The refractive index of the liquids are shown in the adjoining diagram. The container is illuminated from the side and a small piece of glass having refractive index 1.61 is gently dropped into the liquid layer. The glass piece as it descends downwards will not be visible in

    physics-General
    Refractive index of liquid C is same as that of glass piece. So it will not be visible in liquid C.
    General
    maths-

    Values of stretchy integral subscript negative 1 divided by 2 end subscript superscript plus 1 divided by 2 end superscript   c o s invisible function application x l o g invisible function application fraction numerator 1 plus x over denominator 1 minus x end fraction is :

    Values of stretchy integral subscript negative 1 divided by 2 end subscript superscript plus 1 divided by 2 end superscript   c o s invisible function application x l o g invisible function application fraction numerator 1 plus x over denominator 1 minus x end fraction is :

    maths-General
    General
    physics-

    A rectangular tank of depth 8 meter is full of water (mu equals 4 divided by 3), the bottom is seen at the depth

    A rectangular tank of depth 8 meter is full of water (mu equals 4 divided by 3), the bottom is seen at the depth

    physics-General
    General
    chemistry-

    photon of energe12.75eicompletely absorbed by a hydrogeatoinitialliground sate. Thprinciple quantunumber othe excited state is:

    photon of energe12.75eicompletely absorbed by a hydrogeatoinitialliground sate. Thprinciple quantunumber othe excited state is:

    chemistry-General
    General
    chemistry-

    A certain electronic transition from an excited statetothegroundstateoftheHatom in one or morsteps giverise to foulines ithe ultra violet region of the spectrum, how many lines doethitransitioproducithinfrareregion of the spectrum-

    A certain electronic transition from an excited statetothegroundstateoftheHatom in one or morsteps giverise to foulines ithe ultra violet region of the spectrum, how many lines doethitransitioproducithinfrareregion of the spectrum-

    chemistry-General
    General
    chemistry-

    Find ouratio of followinfor photon (νmax.) Lyman: (νmax) Brackett-

    Find ouratio of followinfor photon (νmax.) Lyman: (νmax) Brackett-

    chemistry-General
    General
    chemistry-

    The potential energieofirst, second and third Bohrs orbits of He+ cation arE1,Eand E3. The correcsequence of these energies is-

    The potential energieofirst, second and third Bohrs orbits of He+ cation arE1,Eand E3. The correcsequence of these energies is-

    chemistry-General
    General
    Maths-

    In straight triangle A B C comma left parenthesis a minus b right parenthesis squared cos squared invisible function application c over 2 plus left parenthesis a plus b right parenthesis squared sin squared invisible function application c over 2 equals

    In straight triangle A B C comma left parenthesis a minus b right parenthesis squared cos squared invisible function application c over 2 plus left parenthesis a plus b right parenthesis squared sin squared invisible function application c over 2 equals
    left parenthesis a squared plus b squared minus 2 a b right parenthesis space cos squared C over 2 plus left parenthesis a squared plus b squared plus 2 a b right parenthesis sin squared C over 2
equals a squared cos squared C over 2 plus b squared cos squared C over 2 minus 2 a b cos squared C over 2 plus a squared sin squared C over 2 plus b squared sin squared C over 2 plus 2 a b sin squared C over 2
equals a squared left parenthesis cos squared C over 2 plus sin squared C over 2 right parenthesis plus b squared left parenthesis cos squared C over 2 plus sin squared C over 2 right parenthesis minus 2 a b left parenthesis cos squared C over 2 minus sin squared C over 2 right parenthesis
equals a squared plus b squared minus 2 a b space cos C space space space space space space left parenthesis b y comma cos squared x plus sin squared x equals 1 right parenthesis comma left parenthesis b y comma cos squared x minus sin squared x equals cos 2 x right parenthesis
equals c squared space space space space space space space space space space space space space space space space space space space space left parenthesis b y space cos i n e space f o r m u l a comma space c squared equals a squared plus b squared minus 2 a b space cos C right parenthesis

    In straight triangle A B C comma left parenthesis a minus b right parenthesis squared cos squared invisible function application c over 2 plus left parenthesis a plus b right parenthesis squared sin squared invisible function application c over 2 equals

    Maths-General
    In straight triangle A B C comma left parenthesis a minus b right parenthesis squared cos squared invisible function application c over 2 plus left parenthesis a plus b right parenthesis squared sin squared invisible function application c over 2 equals
    left parenthesis a squared plus b squared minus 2 a b right parenthesis space cos squared C over 2 plus left parenthesis a squared plus b squared plus 2 a b right parenthesis sin squared C over 2
equals a squared cos squared C over 2 plus b squared cos squared C over 2 minus 2 a b cos squared C over 2 plus a squared sin squared C over 2 plus b squared sin squared C over 2 plus 2 a b sin squared C over 2
equals a squared left parenthesis cos squared C over 2 plus sin squared C over 2 right parenthesis plus b squared left parenthesis cos squared C over 2 plus sin squared C over 2 right parenthesis minus 2 a b left parenthesis cos squared C over 2 minus sin squared C over 2 right parenthesis
equals a squared plus b squared minus 2 a b space cos C space space space space space space left parenthesis b y comma cos squared x plus sin squared x equals 1 right parenthesis comma left parenthesis b y comma cos squared x minus sin squared x equals cos 2 x right parenthesis
equals c squared space space space space space space space space space space space space space space space space space space space space left parenthesis b y space cos i n e space f o r m u l a comma space c squared equals a squared plus b squared minus 2 a b space cos C right parenthesis
    General
    maths-

    Tangents drawn to parabola y to the power of 2 end exponent= 4ax at the points A and B intersect at C. Ordinate of A, C and B forms

    Clearly, both the lines pass through (-a, b) which is a point lying on the directrix of the parabola.
    rightwards double arrowC identical to (at subscript 1 end subscript t subscript 2 end subscript, a(t subscript 1 end subscript plus t subscript 2 end subscript))
    Clearly the ordinates of A,C, B are in A.P.

    Tangents drawn to parabola y to the power of 2 end exponent= 4ax at the points A and B intersect at C. Ordinate of A, C and B forms

    maths-General
    Clearly, both the lines pass through (-a, b) which is a point lying on the directrix of the parabola.
    rightwards double arrowC identical to (at subscript 1 end subscript t subscript 2 end subscript, a(t subscript 1 end subscript plus t subscript 2 end subscript))
    Clearly the ordinates of A,C, B are in A.P.