Advertisements
Advertisements
प्रश्न
Solve the triangle in which a = `(sqrt3 + 1)`, b = `(sqrt3 - 1)` and ∠C = 60°.
Advertisements
उत्तर
Given: a = `(sqrt3 + 1)`, b = `(sqrt3 - 1)` and ∠C = 60°
By cosine rule,
c2 = a2 + b2 - 2ab cos C
`= (sqrt3 + 1)^2 + (sqrt3 - 1)^2 - 2(sqrt3 + 1)(sqrt3 - 1) cos 60^circ`
= 3 + 1 + `2sqrt3` + 3 + 1 - `2sqrt3` - 2(3 - 1)`(1/2)`
= 8 - 2 = 6
∴ c = `sqrt6` ....[∵ c > 0]
By sine rule,
`"a"/("sin A") = "b"/("sin B") = "c"/("sin C")`
∴ `(sqrt3 + 1)/("sin A") = (sqrt3 - 1)/("sin B") = sqrt6/("sin" 60^circ)`
∴ `(sqrt3 + 1)/("sin A") = (sqrt3 - 1)/("sin B") = sqrt6/(sqrt3//2) = 2sqrt2`
∴ sin A = `(sqrt3 + 1)/(2sqrt2) and sin "B" = (sqrt3 - 1)/(2sqrt2)`
∴ `sin "A" = sqrt3/(2sqrt2) + 1/(2sqrt2) and sin "B" = sqrt3/(2sqrt2) - 1/(2sqrt2)`
∴ sin A = `sqrt3/2 xx 1/sqrt2 + 1/2 xx 1/sqrt2`
∴ and sin B = `sqrt3/2 xx 1/sqrt2 - 1/2 xx 1/sqrt2`
∴ sin A = sin 60° cos 45° + cos 60° sin 45° and
sin B = sin 60° cos 45° - cos 60° sin 45°
∴ sin A = sin (60° + 45°) = sin 105°
and sin B = sin (60° - 45°) = sin 15°
∴ A = 105° and B = 15°
Hence, A = 105°, B = 15° and C = `sqrt6` units
APPEARS IN
संबंधित प्रश्न
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 ,Δ ABC with usual notations prove that
b2 = c2 +a2 - 2 ca cos B
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))`
Find the polar co-ordinates of the point whose Cartesian co-ordinates are.
`(3/2, (3√3)/2)`.
In any ΔABC, prove the following:
`("c" - "b cos A")/("b" - "c cos A") = ("cos B")/("cos C")`
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, prove that a2 (cos2 B - cos2 C) + b2 (cos2 C - cos2 A) + c2 (cos2 A - cos2 B) = 0.
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 `2 sin^-1 (3/5) = tan^-1(24/7)`
Show that
`tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/3) + tan^-1 (1/8) = pi/4.`
Show that `(9pi)/8 - 9/4 sin^-1 (1/3) = 9/4 sin^-1 ((2sqrt2)/3)`.
In ∆ABC, if b2 + c2 − a2 = bc, then ∠A = ______.
In ∆ABC, if sin2A + sin2B = sin2C, then show that a2 + b2 = c2
Find the Cartesian co-ordinates of point whose polar co-ordinates are `(4, pi/3)`
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 ∆ABC, if ∠A = `pi/2`, then prove that sin(B − C) = `("b"^2 - "c"^2)/("b"^2 + "c"^2)`
In ΔABC, if (a+ b - c)(a + b + c) = 3ab, then ______.
In a ΔABC, c2 sin 2B + b2 sin 2C = ?
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))` = ______
In a ΔABC, 2ab sin`((A + B - C)/2)` = ______
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 ______
In ΔABC, a = 7cm, b = 3cm and c = 8 cm, then angle A is ______
If polar co-ordinates of a point are `(1/2, pi/2)`, then its cartesian co-ordinates are ______.
In `triangleABC,` if a = 3, b = 4, c = 5, then sin 2B = ______.
In ΔABC, if `"a" cos^2 "C"/2 + "c" cos^2 "A"/2 = (3"b")/2`, then a, b, c are in ______.
In a triangle ABC, b = `sqrt3`, c = 1 and ∠A = 30°, then the largest angle of the triangle is ______
If a = 13, b = 14, c = 15, then `cos("A"/2)` = ______.
Find the cartesian co-ordinates of the point whose polar co-ordinates are `(1/2, π/3)`.
In triangle ABC, a = 4, b = 3 and ∠A = 60°. If ' c' is a root of the equation c2 – 3c – k = 0. Then k = ______. (with usual notations)
Let ABC be a triangle such that ∠A = 45°, ∠B = 75° then `"a" + "c"sqrt(2)` is equal to ______. (in usual notation)
In a triangle ABC, in usual notation, (a + b + c)(b + c – a) = λbc will be true if ______.
In ΔABC, `(a - b)^2 cos^2 C/2 + (a + b)^2 sin^2 C/2` is equal to ______.
In ΔABC, a = 3, b = 1, cos(A – B) = `2/9`, find c.
In a triangle ABC with usual notations, if a,b, and c are in arithmetic progression, then, \[\tan\frac{A}{2}\cdot\tan\frac{C}{2}=\]
With usual notations, in a triangle ABC, if θ is any real number, then a cos(B - θ) + b cos (A + θ) is
