Advertisements
Advertisements
Question
Derive Projection formula from Law of cosines
Advertisements
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 Law of cosines.
We have cos A = `("b"^2 + "c"^2 - "a"^2)/(2"bc")`
cos B = `("c"^2 + "a"^2 - "b"^2)/(2"ca")`
cos C = `("a"^2 + "b"^2 - "c"^2)/(2"ab")`
(a) b cos C + cs B
= `"b"(("a"^2 + "b"^2 - "c"^2)/(2"ab")) + "c"(("c"^2 + "a"^2 - "b"^2)/(2"ca"))`
= `("a"^2 + "b"^2 - "c"^2)/(2"a") + ("c"^2 + "a"^2 - "b"^2)/(2"a")`
= `("a" + "b"^2 - "c"^2 + "c"^2 + "a"^2 - "b"^2)/(2"a")`
= `(2"a"^2)/(2"a")`
= a
∴ a = b cos C + c cos B
(b) c cos A + a cos C
= `"c"(("b"^2 + "c"^2 - "a"^2)/(2"bc")) + "a"(("a"^2 + "b"^2 - "c"^2)/(2"ab"))`
= `("b"^2 + "c"^2 - "a"^2)/(2"bc") + ("a"^2 + "b"^2 - "c"^2)/(2"ab")`
= `("b"^2 + "c"^2 - "a"^2 + "a"^2 + "b"^2 - "c"^2)/(2"b")`
= `(2"b"^2)/(2"b")`
= b
∴ b = c cos A + a cos C
(c) c = a cos B + b cos A
= `"a"(("c"^2 + "a"^2 - "b"^2)/(2"ca")) + "b"(("b"^2 + "c"^2 - "a"^2)/(2"bc"))`
= `("c"^2 + "a"^2 - "b"^2)/(2"c") + ("b"^2 + "c"^2 - "a"^2)/(2"c")`
= `("c"^2 + "a"^2 - "b"^2 + "b"^2 + "c"^2 - "a"^2)/(2"c")`
= `(2"c"^2)/(2"c")`
= c
∴ c = a cos B + b cos A
APPEARS IN
RELATED QUESTIONS
In a ∆ABC, if `sin"A"/sin"C" = (sin("A" - "B"))/(sin("B" - "C"))` prove that a2, b2, C2 are in Arithmetic Progression
In a ∆ABC, prove that `sin "B"/sin "C" = ("c" - "a"cos "B")/("b" - "a" cos"C")`
In an ∆ABC, prove that a cos A + b cos B + c cos C = 2a sin B sin C
In a ∆ABC, ∠A = 60°. Prove that b + c = `2"a" cos (("B" - "C")/2)`
In an ∆ABC, prove the following, `"a"sin ("A"/2 + "B") = ("b" + "c") sin "A"/2`
In a ∆ABC, prove the following, a(cos B + cos C) = `2("b" + "c") sin^2 "A"/2`
In a ∆ABC, prove the following, `("a"^2 - "c"^2)/"b"^2 = (sin ("A" - "C"))/(sin("A" + "C"))`
In a ∆ABC, prove the following, `("a"+ "b")/("a" - "b") = tan(("A" + "B")/2) cot(("A" - "B")/2)`
A rope of length 42 m is given. Find the largest area of the triangle formed by this rope and find the dimensions of the triangle so formed
Derive Projection formula from Law of sines
Choose the correct alternative:
In a ∆ABC, if
(i) `sin "A"/2 sin "B"/2 sin "C"/2 > 0`
(ii) sin A sin B sin C > 0 then
A circle touches two of the smaller sides of a ΔABC (a < b < c) and has its centre on the greatest side. Then the radius of the circle is ______.
In an equilateral triangle of side `2sqrt(3)` cm, the circum radius is ______.
If in a ΔABC, the altitudes from the vertices A, B, C on opposite sides are in H.P, then sin A, sin B, sin C are in ______
