Advertisements
Advertisements
Question
With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
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`
In any ΔABC, with usual notations, prove that b2 = c2 + a2 – 2ca cos B.
With usual notations, in ΔABC, prove that a(b cos C − c cos B) = b2 − c2
In , ΔABC prove that
`"sin"(("B" - "C")/2) = (("b" - "c")/"a") "cos"("A"/2)`
In , ΔABC with usual notations prove that
(a-b)2 cos2 `("C"/2) +("a"+"b")^2 "sin"^2("C"/2) = "c"^2`
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 Cartesian co-ordinates of the point whose polar co-ordinates are:
`(1/2, (7pi)/3)`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(sqrt(2), sqrt(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 ΔABC, prove the following:
`("c" - "b cos A")/("b" - "c cos A") = ("cos B")/("cos C")`
In any Δ ABC, prove the following:
a2 sin (B - C) = (b2 - c2) sin A.
In any Δ ABC, prove the following:
`"cos 2A"/"a"^2 - "cos 2B"/"b"^2 = 1/"a"^2 - 1/"b"^2`
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 ΔABC, if `"cos A"/"a" = "cos B"/"b"`, then show that it is an isosceles triangle.
In Δ ABC, if a cos2 `"C"/2 + "c cos"^2 "A"/2 = "3b"/2`, then prove that a, b, c are in A.P.
Show that
`tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/3) + tan^-1 (1/8) = pi/4.`
Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]
Show that `(9pi)/8 - 9/4 sin^-1 (1/3) = 9/4 sin^-1 ((2sqrt2)/3)`.
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 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 b2 + c2 − a2 = bc, then ∠A = ______.
In ∆ABC, prove that ac cos B − bc cos A = a2 − b2
Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1, sqrt(3))`
In ΔABC, a = 3, b = 4 and sin A = `3/4`, find ∠B
With usual notations, prove that `(cos "A")/"a" + (cos "B")/"b" + (cos "C")/"c" = ("a"^2 + "b"^2 + "c"^2)/(2"abc")`
In ∆ABC, if a = 13, b = 14, c = 15, then find the value of cos B
In ΔABC, if a cos A = b cos B, then prove that ΔABC is either a right angled or an isosceles triangle.
In ∆ABC, prove that `(cos 2"A")/"a"^2 - (cos 2"c")/"c"^2 = 1/"a"^2 - 1/"c"^2`
In ∆ABC, if `(2cos "A")/"a" + (cos "B")/"b" + (2cos"C")/"c" = "a"/"bc" + "b"/"ca"`, then show that the triangle is a right angled
In ∆ABC, prove that `sin ((A - B)/2) = ((a - b)/c) cos C/2`
If the angles A, B, C of ΔABC are in A.P. and its sides a, b, c are in G.P., then show that a2, b2, c2 are in A.P.
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, if (a+ b - c)(a + b + c) = 3ab, then ______.
In a ΔABC, cot `(("A - B")/2)* tan (("A + B")/2)` is equal to
In a ΔABC, c2 sin 2B + b2 sin 2C = ?
In a ΔABC if 2 cos C = sin B · cosec A, then ______.
With usual notations, if the angles A, B, C of a Δ ABC are in AP and b : c = `sqrt3 : sqrt2`.
In a triangle ABC, If `(sin "A" - sin "C")/(cos "C" - cos "A")` = cot B, then A, B, C are in ________.
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 Δ ABC; with usual notations, if cos A = `(sin "B")/(sin "C")`, then the triangle is _______.
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 ______.
If P(6, 10, 10), Q(1, 0, -5), R(6, -10, λ) are vertices of a triangle right angled at Q, then value of λ is ______.
In Δ ABC, with the usual notations, if `(tan "A"/2)(tan "B"/2) = 3/4` then a + b = ______.
In ΔABC, if `cosA/a = cosB/b,` then triangle ABC is ______
If cartesian co-ordinates of a point are `(1, -sqrt3)`, then its polar co-ordinates are ______
In ΔABC, a = 7cm, b = 3cm and c = 8 cm, then angle A is ______
In any triangle ABC, the simplified form of `(cos2A)/a^2 - (cos2B)/b^2` is ______
If PQ and PR are the two sides of a triangle, then the angle between them which gives maximum area of the triangle is ______.
If in a `triangle"ABC",` a2cos2 A - b2 - c2 = 0, then ______.
If in ΔABC, `sin "B"/2 sin "C"/2 = sin "A"/2` and 2s is the perimeter of the triangle, then s is ______.
If a = 13, b = 14, c = 15, then `cos("A"/2)` = ______.
In a ΔABC, if `("b" + "c")/11 = ("c" + "a")/12 = ("a" + "b")/13`, then cos C = ______.
In ΔABC with usual notations, if ∠A = 30° and a = 5, then `s/(sumsinA)` is equal to ______.
If in a ΔABC `a cos^2(C/2) + c cos^2(A/2) = (3b)/2`, then the sides a, b and c ______.
In a triangle ABC, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
In ΔABC, `(a - b)^2 cos^2 C/2 + (a + b)^2 sin^2 C/2` is equal to ______.
In any ΔABC, prove that:
(b + c) cos A + (c + a) cos B + (a + b) cos C = a + b + c.
If in ΔABC, `sin A/2 * sin C/2 = sin B/2` and 2s is the perimeter of the triangle, then s = ______.
The perimeter of ΔABC is 20, ∠A = 60°, area of ΔABC = `10sqrt(3)`, then find the values of a, b, c.
In ΔABC, a = 3, b = 1, cos(A – B) = `2/9`, find c.
If the angles A, B, C of a ΔABC are in A.P. and ∠A = 30°, c = 5, then find the values of ‘a’ and ‘b’.
