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Maths-

General

Easy

Question

The angles of a triangle are in the ratio 3:5:10. Then ratio of smallest to greatest side is

$1 : s i n 10^{\circ}$
$1 : 2 s i n 10^{6}$
$1 : c o s 10^{\circ}$
$1 : 2 c o s 10^{\circ}$

hint Hint:

In this question first we have to find the angles of the triangle. Then use the sine rule to find the ratio of largest to smallest side.

The correct answer is: $1 colon 2 c o s invisible function application 10 to the power of ring operator$

Step1: Finding the angles of the triangles.
Let the angles be $3 x comma 5 x$ and $10 x$ .
Let ABC be the triangle with $angle A equals 3 x comma space angle B equals 5 x$ and $angle C equals 10 x$ and the side opposite to $angle A comma angle B comma angle C$ are $a comma b comma c$ respectively.
By using the theorem : Side opposite to greatest angle will be largest
The sum of all the angles of a triangles is $180 degree$
=> $3 x plus 5 x plus 10 x space equals space 180 degree$
=> $x equals 10 degree$
$angle A equals space 3 cross times 10 equals 30 degree angle B equals 5 cross times 10 equals 50 degree angle C equals 10 cross times 10 equals 100 degree$
Step2: Find the ratio of the largest to smallest side.
By using sine rule,
$fraction numerator sin open parentheses A close parentheses over denominator a end fraction equals fraction numerator sin open parentheses B close parentheses over denominator b end fraction equals fraction numerator sin open parentheses C close parentheses over denominator c end fraction$
We have to find the ratio of $c over a$ .
$fraction numerator sin open parentheses 30 degree close parentheses over denominator a end fraction equals fraction numerator sin open parentheses 50 degree close parentheses over denominator b end fraction equals fraction numerator sin open parentheses 100 degree close parentheses over denominator c end fraction$
We know that $sin open parentheses 90 degree plus theta close parentheses equals cos open parentheses theta degree close parentheses$
$sin open parentheses 100 degree close parentheses equals sin open parentheses 90 degree plus 10 degree close parentheses$
=> $cos open parentheses 10 degree close parentheses$
=> $fraction numerator 1 over denominator 2 a end fraction equals fraction numerator sin open parentheses 50 degree close parentheses over denominator b end fraction equals fraction numerator cos open parentheses 10 degree close parentheses over denominator c end fraction$
=> $c over a equals 2 cos open parentheses 10 degree close parentheses$

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