Advertisements
Advertisements
प्रश्न
In a ∆ABC, prove that (a2 – b2 + c2) tan B = (a2 + b2 – c2) tan C
उत्तर
We know `"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
Also we know 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"^2 + "b"^2 - "c"^2)/("a"^2 - "b"^2 + "c"^2) = ("a"^2 + "b"^2 - "c"^2)/("c"^2 + "a"^2 + "b"^2)`
= `(("a"^2 + "b"^2 - "c"^2)/(2"abc"))/(("c"^2 + "a"^2 - "b"^2)/(2"abc"))`
= `(1/"c" xx ("a"^2 + "b"^2 - "c"^2)/(2"ab"))/(1/"b" xx ("c"^2 + "a"^2 - "b"^2)/(2"ca"))`
= `(1/"c" xx cos "C")/(1/"b" xx cos "B")`
= `("b" cos "C")/("c" cos "B")`
= `(2"R" sin"B" cos"C")/(2"R" sin"C" cos"B")`
`("a"^2 + "b"^2 - "c"^2)/("a"^2 - "b"^2 + "c"^2) = sin"B"/sin"C" * cos"C"/sin"C"`
`("a"^2 + "b"^2 - "c"^2)/("a"^2 - "b"^2 + "c"^2)` = tan B . cot C
`("a"^2 + "B"^2 - "c"^2)/cot"C"` = (a2 – b2 + c2) tan B
(a2 + b2 + c2) tan C = (a2 – b2 + c2) tan B
APPEARS IN
संबंधित प्रश्न
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 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"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
Derive Projection formula from Law of cosines
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 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 ______
