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In a triangle PQR, i f \angle R= \frac{\pi }{2}.If \tan \left ( \frac{P}{2} \right )\: and \: \tan \left ( \frac{Q}{2} \right )  are the root of ax^{2}+bx+c =0,a\neq 0

  • Option 1)

    b=a+c

  • Option 2)

    b=c

  • Option 3)

    c=a+b

  • Option 4)

    a=b+c

 

Answers (1)

best_answer

As we learnt in 

Trigonometric Equations -

The equations involving trigonometric function of unknown angles are known as trigonometric equations.

- wherein

e.g. \cos ^{2}\Theta - 4\cos \Theta = 1

 

 Sum of roots  = \frac{-b}{a} ..................\left ( 1 \right )

\Rightarrow \tan \frac{p}{2} +\tan \frac{q}{2} = \frac{-b}{a}

\Rightarrow \frac{sin\frac{p}{2}}{cos \frac{p}{2}} +\frac{sin \frac{q}{2}}{cos\frac{q}{2}} =\frac{-b}{a}  \Rightarrow \frac{sin\frac{p}{2} cos\frac{q}{2}+sin\frac{q}{2} cos\frac{p}{2}}{cos\frac{p}{2} cos\frac{q}{2}}=\frac{-b}{a}

\Rightarrow \frac{sin\frac{p+q}{2}}{cos\frac{p}{2} cos\frac{q}{2}} =\frac{-b}{a}    \Rightarrow \frac{sin\frac{\pi -R}{2}}{cos\frac{p}{2}cos\frac{q}{2}} =\frac{-b}{a}

\Rightarrow \frac{sin\frac{\pi }{4}}{cos\frac{p}{2}cos\frac{q}{2}} =\frac{-b}{a}----\left ( 2 \right )                        \left [ \therefore R=\frac{\pi }{2} \right ]

\Rightarrow cos\frac{p}{2}cos\frac{q}{2}=\frac{-a}{b} sin \frac{\pi }{4}.......\left ( 2a \right )

Similarly, product  of roots = \frac{c}{a}----------\left ( 2 \right )

tan \frac{p}{2} tan\frac{\Theta }{2}= \frac{c}{a}   \Rightarrow \frac{sin \frac{p}{2}sin\frac{q}{2}}{cos\frac{p}{2}cos\frac{q}{2}}=\frac{c}{a}..................\left ( 3 \right )

Divides (2) and (3) we get

\frac{sin\frac{\pi }{4}}{sin\frac{p}{2}sin\frac{q}{2}} = \frac{-b}{c}   \Rightarrow sin\frac{p}{2}sin \frac{q}{2} = \frac{-c}{b}sin\frac{\pi }{4}........\left ( 5 \right )

Now (2a) - (5) gives, we get

cos \frac{p+q}{2}=\left [ \frac{-a}{b}+\frac{c}{b} \right ]sin \frac{\pi }{4}    \Rightarrow cos\frac{\pi }{4} = \left ( \frac{-a}{b}+\frac{c}{b} \right )sin \frac{\pi }{4}

\Rightarrow b+a=c


Option 1)

b=a+c

Incorrect

Option 2)

b=c

Incorrect

Option 3)

c=a+b

Correct

Option 4)

a=b+c

Incorrect

Posted by

Aadil

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