Math, asked by janab2, 11 months ago

MOST POPULAR QUESTION:
given that a \: sinB + b \: sinB = c \:
Please prove that:
 = > ({a \: cosB - b \: sinB}) = \sqrt{ {a}^{2} + {b}^{2} - {c}^{2}.}

Answers

Answered by kritanshu
4

Solution:

It is given that a \: sinB + b \: sinB = c \:

a \: sinB + b \: sinB = c \: (given)

Squaring both sides,

So, \: {(a \: sinB + b \: sinB) }^{2} = {c}^{2} .

 = > {a}^{2} {sin}^{2} B + {b}^{2} {sin}^{2} B + 2ab \: sinB \: cosB = {c}^{2}

 = > {a}^{2} (1 - {cos}^{2} B) + {b}^{2} (1 - {sin}^{2} B) + 2ab \: sinB \: cosB = {c}^{2}

 = > {a}^{2} - {a}^{2} {cos}^{2} + {b}^{2}\: - {b}^{2} {sin}^{2} B + 2ab \: sinB \: cosB = {c}^{2}

 = > {a}^{2} {cos}^{2} B - 2ab \: sinB \: cosB \: + {b}^{2} {sin}^{2} B= {a}^{2} + {b}^{2} - {c}^{2}. \:

 = > ({a \: cosB - b \: sinB})^{2} = {a}^{2} + {b}^{2} - {c}^{2}

 = > ({a \: cosB - b \: sinB}) = \sqrt{ {a}^{2} + {b}^{2} - {c}^{2}.}

Hence, it is proved.


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tanusagar91131: no no
Answered by Anonymous
3

\huge\tt{AnsweR}

Squaring both sides:

\sf ={(a \: sinB + b \: sinB) }^{2} = {c}^{2}

\sf ={a}^{2} {sin}^{2} B + {b}^{2} {sin}^{2} B + 2ab \: sinB \: cosB = {c}^{2}

\sf ={a}^{2} (1 - {cos}^{2} B) + {b}^{2} (1 - {sin}^{2} B) + 2ab \: sinB \: cosB = {c}^{2}

\sf ={a}^{2} - {a}^{2} {cos}^{2} + {b}^{2}\: - {b}^{2} {sin}^{2} B + 2ab \: sinB \: cosB = {c}^{2}

\sf ={a}^{2} {cos}^{2} B - 2ab \: sinB \: cosB \: + {b}^{2} {sin}^{2} B= {a}^{2} + {b}^{2} - {c}^{2}

\sf =({a \: cosB - b \: sinB})^{2} = {a}^{2} + {b}^{2} - {c}^{2}

\sf =({a \: cosB - b \: sinB}) = \sqrt{ {a}^{2} + {b}^{2} - {c}^{2}}

Therefore Proved!

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