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Maths-

General

Easy

Question

The perimeter of a Δ $A B C space$ is 6 times the A.M of the sines of its angles if the side ' is , then the angle is

$π / 6$
$π / 3$
$π / 2$
$π$

hint Hint:

We will use the sine rule to find the value of the angle.
$fraction numerator a over denominator sin open parentheses A close parentheses end fraction equals fraction numerator b over denominator sin open parentheses B close parentheses end fraction equals fraction numerator c over denominator sin open parentheses C close parentheses end fraction equals lambda$ . First we will find $lambda$ from the given conditions and then we will find the angle A

The correct answer is: $pi divided by 6$

In this question we are given that the perimeter of a triangle is $6$ times the Arithmetic mean of the $sin e s$ of it's angles.
Let ABC be a triangles with angles $angle A comma angle B comma angle C$ and the sides opposite to angles $angle A comma angle B comma angle C$ as $a comma b comma c$ respectively.
$a plus b plus c equals fraction numerator 6 left parenthesis sin open parentheses A close parentheses plus sin open parentheses B close parentheses plus sin open parentheses C close parentheses right parenthesis over denominator 3 end fraction$
Step1: By using the sine rule we get
$fraction numerator a over denominator sin open parentheses A close parentheses end fraction equals fraction numerator b over denominator sin open parentheses B close parentheses end fraction equals fraction numerator c over denominator sin open parentheses C close parentheses end fraction equals lambda$
=> $lambda left parenthesis sin open parentheses A close parentheses plus sin open parentheses B close parentheses plus sin open parentheses C close parentheses right parenthesis equals fraction numerator 6 left parenthesis sin open parentheses A close parentheses plus sin open parentheses B close parentheses plus sin open parentheses C close parentheses right parenthesis over denominator 3 end fraction$
=> $lambda equals 6 over 2$
=> $lambda equals 2$
Step2: Finding the angle A.
$fraction numerator a over denominator sin open parentheses A close parentheses end fraction equals lambda$
We are given $a equals 1$ and we have calculated $lambda equals 2$
$sin open parentheses A close parentheses equals 1 half$
$angle A equals pi over 6$

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