Advertisements
Advertisements
प्रश्न
Derive Projection formula from Law of cosines
उत्तर
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
संबंधित प्रश्न
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, if cos C = `sin "A"/(2sin"B")` show that the triangle is isosceles
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"+ "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 a ΔABC, let BC = 3. D is a point on BC such that BD = 2, Then the value of AB2 + 2AC2 – 3AD2 is ______.
In an equilateral triangle of side `2sqrt(3)` cm, the circum radius is ______.
Let a, b and c be the length of sides of a triangle ABC such that `(a + b)/7 = (b + c)/8 = (c + a)/9`. If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of `R/r` is equal to ______.
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 ______
