Advertisements
Advertisements
Question
With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
Advertisements
Solution
By sine rule,
`"a"/("sin A") = "b"/("sin B") = "c"/("sin C")` = k
`"a"/("sin A") = "k", "b"/("sin B") = "k", "c"/("sin C")` = k
∴ sin A = `"a"/"k"`, sin B = `"b"/"k"`, sin C = `"c"/"k"`
Now,
(c2 − a2 + b2) tan A = (c2 − a2 + b2). `"sin A"/"cos A"`
`= ("c"^2 + "b"^2 - "a"^2) xx ("k"/"a")/((("c"^2 + "b"^2 - "a"^2)/"2bc"))`
`= ("c"^2 + "b"^2 - "a"^2)/"k" xx "2abc"/("c"^2 + "b"^2 - "a"^2)`
= `("2abc")/"k"` .....(1)
(a2 - b2 + c2) tan B = (a2 - b2 + c2) .`"sin B"/"cos B"`
`= ("a"^2 + "c"^2 - "b"^2) xx ("k"/"b")/((("a"^2 + "c"^2 - "b"^2)/"2ac"))`
`= ("a"^2 + "c"^2 - "b"^2)/"k" xx "2abc"/("a"^2 + "c"^2 - "b"^2)`
= `("2abc")/"k"` ....(2)
(b2 − c2 + a2) tan C = (b2 − c2 + a2). `"sin C"/"cos C"`
`= ("a"^2 + "b"^2 - "c"^2) xx ("k"/"c")/((("a"^2 + "b"^2 - "c"^2)/"2ab"))`
`= ("a"^2 + "b"^2 - "c"^2)/"k" xx "2abc"/("a"^2 + "b"^2 - "c"^2)`
= `("2abc")/"k"` .....(3)
From (1), (2) and (3), we get
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
`("2abc")/"k" = ("2abc")/"k" = ("2abc")/"k"`
1 = 1= 1
All are equals.
L. H. S. = R. H. S.
APPEARS IN
RELATED QUESTIONS
In any ΔABC if a2 , b2 , c2 are in arithmetic progression, then prove that Cot A, Cot B, Cot C are in arithmetic progression.
In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot C/2`.
If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to
(A) `1/sqrt5`
(B) `1/sqrt10`
(C) `1/sqrt15`
(D) `1/(2sqrt5)`
With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(sqrt(2), pi/4)`
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(3/4, (3pi)/4)`
Find the polar coordinates of the point whose Cartesian coordinates are `(1, - sqrt(3))`.
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(3/2, (3√3)/2)`.
Solve the triangle in which a = `(sqrt3 + 1)`, b = `(sqrt3 - 1)` and ∠C = 60°.
In any Δ ABC, prove the following:
a sin A - b sin B = c sin (A - B)
In any Δ ABC, prove the following:
a2 sin (B - C) = (b2 - c2) sin A.
In any Δ ABC, prove the following:
ac cos B - bc cos A = a2 - b2
In Δ ABC, if a, b, c are in A.P., then show that cot `"A"/2, cot "B"/2, cot "C"/2` are also in A.P.
In Δ ABC, if ∠C = 90°, then prove that sin (A - B) = `("a"^2 - "b"^2)/("a"^2 + "b"^2)`
In ΔABC, if `"cos A"/"a" = "cos B"/"b"`, then show that it is an isosceles triangle.
In Δ ABC, prove that a2 (cos2 B - cos2 C) + b2 (cos2 C - cos2 A) + c2 (cos2 A - cos2 B) = 0.
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.
In ∆ABC, if cos A = `(sinB)/(2sinC)`, then ∆ABC is ______.
In ∆ABC, if ∠A = 30°, ∠B = 60°, then the ratio of sides is ______.
In ∆ABC, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a2
In ∆ABC, if a = 13, b = 14, c = 15, then find the value of cos B
In ∆ABC, prove that `sin ((A - B)/2) = ((a - b)/c) cos C/2`
In ΔABC, prove that `("b"^2 - "c"^2)/"a" cos"A" + ("c"^2 - "a"^2)/"b" cos"B" + ("a"^2 - "b"^2)/"c" cos "C"` = 0
In ΔABC, a(cos2B + cos2C) + cos A(c cos C + b cos B) = ?
In ΔABC, if (a+ b - c)(a + b + c) = 3ab, then ______.
In a triangle ABC with usual notations, if `(cos "A")/"a" = (cos "B")/"b" = (cos "C")/"c"`, then area of triangle ABC with a = `sqrt6` is ____________.
If in a right-angled triangle ABC, the hypotenuse AB = p, then `overline"AB".overline" AC" + overline"BC".overline" BA" + overline" CA".overline"CB"` is equal to ______
In a ΔABC, `(sin "C"/2)/(cos(("A" - "B")/2))` = ______
If `(- sqrt2, sqrt2)` are cartesian co-ordinates of the point, then its polar co-ordinates are ______.
In Δ ABC; with usual notations, `("b" sin "B" - "c" sin "C")/(sin ("B - C"))` = _______.
The polar co-ordinates of P are `(2, pi/6)`. If Q is the image of P about the X-axis then the polar co-ordinates of Q are ______.
In ΔABC, if `cosA/a = cosB/b,` then triangle ABC is ______
In any triangle ABC, the simplified form of `(cos2A)/a^2 - (cos2B)/b^2` is ______
The smallest angle of the ΔABC, when a = 7, b = `4sqrt(3)` and c = `sqrt(13)` is ______.
In `triangleABC,` if a = 3, b = 4, c = 5, then sin 2B = ______.
If in ΔABC, `sin "B"/2 sin "C"/2 = sin "A"/2` and 2s is the perimeter of the triangle, then s is ______.
In a triangle ABC, b = `sqrt3`, c = 1 and ∠A = 30°, then the largest angle of the triangle is ______
In ΔABC, `cos"A"/"a" = cos"B"/"b" cos"C"/"c"`. If a = `1/sqrt(6)`, then the area of the triangle is ______.
If in a triangle ABC, AB = 5 units, AB = 5 units, ∠B = `cos^-1 (3/5)` and radius of circumcircle of ΔABC is 5 units, then the area (in sq.units) of ΔABC is ______.
The number of solutions of the equation sin 2x – 2 cosx + 4 sinx = 4 in the interval [0, 5π] is ______.
In a triangle ABC, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
In ΔABC, a = 3, b = 1, cos(A – B) = `2/9`, find c.
