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Question

If the angles of a triangle are in the ratio 1:3:5 and θdenote the smallest angle, then the ratio of the largest side to the smallest side of the triangle is:


A

(3sinθ+cosθ)2sinθ

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B

(3cosθ-sinθ)2sinθ

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C

(cosθ+3sinθ)2sinθ

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D

(3cosθ+sinθ)2sinθ

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Solution

The correct option is D

(3cosθ+sinθ)2sinθ


Explanation for the correct option.

Step 1: Find the value of angles

Given that, the angles of a triangle are in the ratio 1:3:5 & θ is the smallest angle.

Let angles be A,B,C.

Then, A=x,B=3x,C=5x

Now using the angle sum property of a triangle I.e; the sum of all angles in a triangle is 180ο.

x+3x+5x=180ο9x=180οx=20ο

Therefore, A=20ο,B=60ο,C=100ο

Step 2: Find the ratio of the largest side to the smallest side of the triangle

Now use the sine rule i.e; asinA=bsinB=csinC.

Since here angle C>angleA so we need to find ca.

ca=sinCsinA=sin100οsin20ο=sin120ο-θsin20ο=(sin120οcos20ο-cos120οsin20ο)sin20ο=32cos20ο+12sin20οsin20ο=3cos20ο+sin20ο2sin20ο=3cosθ+sinθ2sinθ

Hence, option D is correct.


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