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Question
Derive Projection formula from Law of sines
Solution
To prove
(a) a = b cos C + c cos B
(b) b = c cos A + a cos C
(c) c = a cos B + b cos A
Using the Law of sines,
We know that `"a"/sin"A" = "b"/sin "B" = "c"/sin "C"` = 2R
`"a"/sin"A"` = 2R ⇒ a = 2R sin A
`"b"/sin"B"` = 2R ⇒ b = 2R sin B
`"c"/sin"C"` = 2R ⇒ c = 2R sin C
(a) b cos C + c cos B = 2 R sin B cos C + 2 R sin C cos B
= 2R (sin B cos C + cos B sin C)
= 2R sin (B + C)
= 2R sin (180° – A)
b cos C + c cos B = 2R sin A = a
a = b cos C + c cos B
(b) c cos A + a cos C = 2R sin C cos A + 2R sin A cos C
= 2R (sin C cos A + cos C sin A)
= 2R sin(C + A)
= 2R sin(180° – B)
= 2R sin B = b
∴ b = c cos A + a cos C
(c) a cos B + b cosA = 2R sin A cos B + 2R sin B cos A
= 2R (sin A cos B + cos A sin B)
= 2R sin(A + B)
= 2R sin(180° – C)
= 2R sin C = c
∴ c = a cos B + b cos A
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