Explanation
To find the length of a chord (L) that subtends an angle θ at the center of a circle with radius r, we use the formula:
L=2rsin(2θ)
Step 1: Identify given values
Radius of the unit circle, r=1
Angle subtended at the center, θ=45∘
Step 2: Apply the formula
L=2(1)sin(245∘)=2sin(22.5∘)
Step 3: Calculate sin(22.5∘) using the half-angle formula
We know that sin(2α)=21−cosα. Setting α=45∘:
sin(22.5∘)=21−cos45∘=21−21=222−1=222−1
To simplify, multiply the numerator and denominator by 2:
sin(22.5∘)=42−2=22−2
Step 4: Final Calculation
L=2⋅(22−2)=2−2
Final Answer:
The length of the chord is 2−2 units. The correct option is (d).
