In the diagram below, B C is the diameter and O D is the radius of the semicircle centered at O . If A D = D C , what is sin ∠ O A C ?
Give your answer correct to three decimal places.
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O D = O C ⟹ ∡ O D C = ∡ B A C = 4 5 ∘
A B = B C = 2 × B O
∡ B A C = arctan 2 1 ≈ 2 6 . 5 6 5 ∘
∡ O A C = 4 5 ∘ − 2 6 . 5 6 5 ∘ = 1 8 . 4 3 5 ∘
sin O A C = sin ( 1 8 . 4 3 5 ∘ ) = 0 . 3 1 6
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We can calculate the answer without using calculator until the final answer, here is a way:
Let O E = a , O E is drawn such that O E ⊥ D C
∴ O C = 2 a , B O = O C = 2 a
∴ A B = 2 2 a
∴ A O = ( 2 2 a ) 2 + ( 2 a ) 2 = 1 0 a
∴ s i n ∠ O A C = A O O E = 1 0 a a = 1 0 1 0 = 0 . 3 1 6