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Maths-

General

Easy

Question

In $increment A B C comma$ the median $A D$ divides $angle B A C$ such that $angle B A D colon angle C A D equals 2 colon 1.$ Then $cos invisible function application left parenthesis A divided by 3 right parenthesis blank$ is equal to

$\frac{\sin B}{2 \sin C}$
$\frac{\sin C}{2 \sin B}$
$\frac{2 \sin B}{\sin C}$
None of these

The correct answer is: $fraction numerator sin invisible function application B over denominator 2 sin invisible function application C end fraction$

Let $blank fraction numerator A over denominator 3 end fraction equals angle C A D equals theta$
Now, by $m minus n$ theorem,
$open parentheses 1 plus 1 close parentheses cot invisible function application alpha equals 1 cot invisible function application 2 theta minus 1 cot invisible function application theta blank rightwards double arrow 2 cot invisible function application open parentheses B plus 2 theta close parentheses equals cot invisible function application 2 theta minus cot invisible function application theta$
$rightwards double arrow cot invisible function application open parentheses B plus 2 theta close parentheses plus cot invisible function application theta equals cot invisible function application 2 theta minus cot invisible function application open parentheses B plus 2 theta close parentheses$
$rightwards double arrow fraction numerator sin invisible function application open parentheses B plus 3 theta close parentheses over denominator sin invisible function application open parentheses B plus 2 theta close parentheses sin invisible function application theta end fraction equals fraction numerator sin invisible function application B over denominator sin invisible function application open parentheses B plus 2 theta close parentheses sin invisible function application 2 theta end fraction$
$rightwards double arrow fraction numerator sin invisible function application open parentheses B plus A close parentheses over denominator sin invisible function application theta end fraction equals fraction numerator sin invisible function application B over denominator sin invisible function application 2 theta end fraction$
$rightwards double arrow sin invisible function application C equals fraction numerator sin invisible function application B over denominator 2 cos invisible function application theta end fraction$
$rightwards double arrow cos invisible function application fraction numerator A over denominator 3 end fraction equals fraction numerator sin invisible function application B over denominator 2 sin invisible function application C end fraction$

Related Questions to study

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The number of values of $y blank$ in $blank open square brackets negative 2 pi comma blank 2 pi close square brackets$ satisfying the equation $open vertical bar sin invisible function application 2 x close vertical bar plus open vertical bar cos invisible function application 2 x close vertical bar equals open vertical bar sin invisible function application y close vertical bar$ is

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