Question

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■ 20 POINTS
● FULL EXPLAINATION
◆ NO USELESS ANSWER
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→ IF 2Cosx + Sinx =1 , then value of 7cosx + 6sinx is equals to

A) 2 OR 6

B) 1 OR 3

C) 2 OR 3

D) NONE OF THESE
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FULL EXPLAINATION :-)

Answer 1

Cosx = (1-tan2x/2)/(1+tan2x/2)

Sin x = (2tanx/2)/(1+tan2x/2)

2cosx + sinx=1

2(((1-tan2x/2)/(1+tan2x/2)) + ((2tanx/2)/(1+tan2x/2))=1

Solving this we get quadratic in tanx/2

3tan2x/2 – 2tanx/2 -1=0

(3tanx/2+1)(tanx/2 -1)=0

Tanx/2 =1 or -1/3

7cosx + 6sinx

7((1-tan2x/2)/(1+tan2x/2)) + 6((2tanx/2)/(1+tan2x/2))

Putting value of Tanx/2, we get

When

Tanx/2 =1 Ans=6

When

Tanx/2=-1/3 Ans=2


A is the correct option.

Answer 2

Answer:

Option (A)

Explanation:

Given Equation is 2cosx + sinx = 1.

= > 2cosx = 1 - sinx

On squaring both sides, we get

= > (2cosx)^2 = (1 - sinx)^2

= > 4cos^2x = 1 + sin^2x - 2sinx

= > 4(1 - sin^2x) = 1 + sin^2x - 2sinx

= > 4 - 4sin^2x = 1 + sin^2x - 2sinx

= > 1 + sin^2x - 2sinx - 4 + 4sin^2x = 0

= > 5sin^2x - 2sinx - 3 = 0

= > 5sin^2x - 5sinx + 3sinx - 3 = 0

= > 5sin^2x + 3sinx - 5sinx - 3 = 0

= > sinx(5sinx + 3) - 1(5sinx + 3) = 0

= > (sinx - 1)(5sinx + 3) = 0

= > (sinx - 1) = 0 , (5sinx + 3) = 0

sinx = 1, sinx = -3/5


(1) When sinx = 1 and cosx = 0: 

= > 7cosx + 6 sinx

= > 7(0) + 6(1)

= > 6.



(2) when cosx = 0 and sinx = -3/5:

= > 7 cosx + 6sinx

$= \ \textgreater \ 7( \sqrt{1 - sin^2x} ) + 6sinx$

$= \ \textgreater \ 7( \sqrt{1 - ( \frac{9}{25} } ) + 6( \frac{-3}{5})$

$= \ \textgreater \ 7( \sqrt{ \frac{25 - 9}{25} } ) + \frac{-18}{5}$

$= \ \textgreater \ 7( \sqrt{ \frac{16}{25} } ) - \frac{18}{5}$

$= \ \textgreater \ 7( \frac{4}{5} ) - \frac{18}{5}$

$= \ \textgreater \ \frac{28}{5} - \frac{18}{5}$

$= \ \textgreater \ \frac{10}{5}$

= > 2.



Therefore 7cosx + 6sinx = 2 (or) 6.


Hope this helps!