Advertisements
Advertisements
प्रश्न
In a ∆ABC, prove that `sin "B"/sin "C" = ("c" - "a"cos "B")/("b" - "a" cos"C")`
उत्तर
We have `"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
cos B = `("c"^2 + "a"^2 - "b"^2)/(2"ca")`,
cos C = `("a"^2 + "b"^2 - "c"^2)/(2"ab")`
`("c"- "a" cos "B")/("b" - "a" cos "C") = ("c" - "a"(("c"^2 + "a"^2 - "b"^2)/(2"ca")))/("b"- "a" (("a"^2 + "b"^2 - "c"^2)/(2"ab"))`
= `("c" - (("c"^2 + "a"^2 - "b"^2)/(2"c")))/("b" - (("a"^2 + "b"^2 - "c"^2)/(2"b"))`
= `((2"c"^2 - ("c"^2 + "a"^2 - "b"^2))/(2"c"))/((2"b"^2 - ("a"^2 + "b"^2 - ""^2))/(2"b"))`
= `(2"c"^2 - "c"^2 - "a"^2 + ""^2)/(2"b"^2 - "a"^2 -"b"^2 + "c"^2) xx"b"/"c"`
= `("c"^2 + "b"^2 - "a"^2)/("b"^2 + "c"^2 - "a"^2) xx "b"/"c"`
=`"b"/"c"`
= `(2"R" sin"B")/(2"R" sin"C")`
`("c" - "a"cos "B")/("b" - "a" cos"C") = sin"B"/sin"C"`
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
The angles of a triangle ABC, are in Arithmetic Progression and if b : c = `sqrt(3) : sqrt(2)`, find ∠A
In a ∆ABC, if cos C = `sin "A"/(2sin"B")` show that the triangle is isosceles
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 a ∆ABC, prove the following, `("a"sin("B" - "C"))/("b"^2 - "c"^2) = ("b"sin("C" - "A"))/("c"^2 - "a"^2) = ("c"sin("A" - "B"))/("a"^2 - "b"^2)`
In a ∆ABC, prove the following, `("a"+ "b")/("a" - "b") = tan(("A" + "B")/2) cot(("A" - "B")/2)`
An Engineer has to develop a triangular shaped park with a perimeter 120 m in a village. The park to be developed must be of maximum area. Find out the dimensions of the park
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
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 usual notation a ΔABC, if A, A1, A2, A3 be the area of the in-circle and ex-circles, then `1/sqrt(A_1) + 1/sqrt(A_2) + 1/sqrt(A_3)` is equal to ______.
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 ______
