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Maths-

General

Easy

Question

In triangle $A B C$ , angle $A$ is greater than angle $B$ . If the measures of angles $A$ and $B$ satisfy the equation $3 sin invisible function application x minus 4 sin to the power of 3 end exponent invisible function application x minus k equals 0 comma 0 less than k less than 1$ , then the measure of angle $C$ is

$\frac{π}{3}$
$\frac{π}{2}$
$\frac{2 π}{3}$
$\frac{5 π}{6}$

The correct answer is: $fraction numerator 2 pi over denominator 3 end fraction$

Given that $A greater than B$
and $3 sin invisible function application x minus 4 sin to the power of 3 end exponent invisible function application x minus k equals 0 comma blank 0 less than k less than 1 blank rightwards double arrow sin invisible function application 3 x equals k$
As $A$ and $B$ satisfy above equation [given]
$therefore sin invisible function application 3 A equals k comma sin invisible function application 3 B equals k blank rightwards double arrow sin invisible function application 3 A minus sin invisible function application 3 B equals 0$
$rightwards double arrow 2 cos invisible function application fraction numerator 3 A plus 3 B over denominator 2 end fraction sin invisible function application fraction numerator 3 A minus 3 B over denominator 2 end fraction equals 0$
$rightwards double arrow cos invisible function application open parentheses fraction numerator 3 A plus 3 B over denominator 2 end fraction close parentheses equals 0 blank o r sin invisible function application open parentheses fraction numerator 3 A minus 3 B over denominator 2 end fraction close parentheses equals 0$
$rightwards double arrow fraction numerator 3 A plus 3 B over denominator 2 end fraction equals 90 degree blank o r blank fraction numerator 3 A minus 3 B over denominator 2 end fraction equals 0$
$rightwards double arrow A plus B equals 60 degree$ or $A equals B$
But given that $A greater than B$ , therefore $A not equal to B$
Thus, $A plus B equals 60 degree$
But $A plus B plus C equals 180 degree$
$therefore C equals 180 degree minus 60 degree equals 120 degree$
$equals 2 pi divided by 3$

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