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Maths-

General

Easy

Question

If $a sin space x plus b cos space left parenthesis x plus theta right parenthesis plus b cos space left parenthesis x minus theta right parenthesis equals$ d then the minimum value of $vertical line cos space theta vertical line$ is equal to (where is variable)

$\frac{1}{2 | b |} \sqrt{d^{2} - a^{2}}$
$\frac{1}{2 | a |} \sqrt{d^{2} - a^{2}}$
$\frac{1}{2 | d |} \sqrt{d^{2} - a^{2}}$
$\frac{1}{| d |} \sqrt{d^{2} - a^{2}}$

hint Hint:

In the question we will first use the formula cos(A+B)=cosA.cosB-sinA.sinB and cos(A-B)=cosA.cosB+sinA.sinB then we will convert sin and cos in terms of tan using the formula $sin 2 A equals fraction numerator 2 tan A over denominator 1 plus tan squared A end fraction a n d space cos 2 A equals fraction numerator 1 minus tan squared A over denominator 1 plus tan squared A end fraction$ then we will make a quadratic equation assuming x in terms of tan. Then we will use the concept of D is greater than 0 to get the required result.

The correct answer is: $fraction numerator 1 over denominator 2 vertical line b vertical line end fraction square root of d squared minus a squared end root$

$a sin space x plus b cos space left parenthesis x plus theta right parenthesis plus b cos space left parenthesis x minus theta right parenthesis equals d rightwards double arrow a sin x plus b open parentheses cos x space cos theta minus sin x space sin theta close parentheses plus b open parentheses cos x space cos plus sin x space sin theta close parentheses equals d rightwards double arrow a space sin x plus b cos x space cos theta minus b sin x space sin theta plus b cos x space cos plus b sin x space sin theta equals d rightwards double arrow a sin x space plus 2 b. cos x. cos theta equals d rightwards double arrow a open parentheses fraction numerator 2 tan begin display style x over 2 end style over denominator 1 plus tan squared begin display style x over 2 end style end fraction close parentheses plus 2 b. open parentheses fraction numerator 1 minus tan squared begin display style x over 2 end style over denominator 1 plus tan squared begin display style x over 2 end style end fraction close parentheses. cos theta equals d rightwards double arrow 2 a space tan x over 2 plus 2 b open parentheses 1 minus tan squared x over 2 close parentheses. cos theta equals d open parentheses 1 plus tan squared x over 2 close parentheses rightwards double arrow 2 a. tan x over 2 plus 2 b cos theta minus 2 b space tan squared x over 2 cos theta minus d minus d tan squared x over 2 equals 0 rightwards double arrow negative tan squared x over 2 open parentheses 2 b cos theta plus d close parentheses plus 2 a. tan x over 2 plus 2 b cos theta minus d equals 0 rightwards double arrow tan squared x over 2 open parentheses 2 b cos theta plus d close parentheses minus 2 a. tan x over 2 minus 2 b cos theta plus d equals 0 w e space k n o w space t h a t space space D greater or equal than 0 comma b squared minus 4 a c greater or equal than 0 open parentheses negative 2 a close parentheses squared minus 4 open parentheses 2 b cos theta plus d close parentheses open parentheses d minus 2 b cos theta close parentheses greater or equal than 0 4 a squared minus 4 open square brackets d squared minus open parentheses 2 b cos theta close parentheses squared close square brackets greater or equal than 0 4 open parentheses a squared minus d squared plus 4 b squared cos squared theta close parentheses greater or equal than 0 4 b squared cos squared theta greater or equal than d squared minus a squared cos squared theta greater or equal than fraction numerator d squared minus a squared over denominator 4 b squared end fraction open vertical bar cos theta close vertical bar greater or equal than square root of fraction numerator d squared minus a squared over denominator 4 b squared end fraction end root open vertical bar cos theta close vertical bar greater or equal than fraction numerator square root of d squared minus a squared end root over denominator 2 open vertical bar b close vertical bar end fraction S o comma space t h e space m i n i m u m space v a l u e space o f open vertical bar cos theta close vertical bar equals fraction numerator square root of d squared minus a squared end root over denominator 2 open vertical bar b close vertical bar end fraction$

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