UPSC CDS 1 2022 Math Paper

Given:
The radius of the circle is 1 unit
Angle subtends by the chord at the center =2θ
Concept Used:
Sinθ= Perpendicular/Hypotenuse
Calculation:Construction: Draw a line OM⟂AB
Now, In △AOM and △BOM
OM=OM‌‌[ common side ]
∠AMO=∠BMO=90∘
And OA=OB [Hypotanuse of triangles]
So, △AOM≅△BOM by R-H-S Rule of congruency
So, ∠AOM=∠BOM=2θ∕2=θ
And AM=BM=AB∕2 . . . (i)
Now, In △BOM
sin‌θ=BM∕OB=BM∕1
⇒BM=sin‌θ
Now, from (i), we get
⇒AB∕2=sin‌θ
⇒AB=2‌sin‌θ
∴ The length of chord is 2‌sin‌θ .