Advertisements
Advertisements
Question
Solve the triangle in which a = `(sqrt3 + 1)`, b = `(sqrt3 - 1)` and ∠C = 60°.
Advertisements
Solution
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
RELATED QUESTIONS
In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a + c - b)
The angles of the ΔABC are in A.P. and b:c=`sqrt3:sqrt2` then find`angleA,angleB,angleC`
In ,Δ ABC with usual notations prove that
b2 = c2 +a2 - 2 ca cos B
Find the Cartesian coordinates of the point whose polar coordinates are :
`(4, pi/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 any Δ ABC, prove the following:
ac cos B - bc cos A = a2 - b2
With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
Show that `2 sin^-1 (3/5) = tan^-1(24/7)`
Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]
If sin `(sin^-1 1/5 + cos^-1 x) = 1`, then find the value of x.
If `tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4`, find the value of x.
State whether the following equation has a solution or not?
cos 2θ = `1/3`
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, if b2 + c2 − a2 = bc, then ∠A = ______.
If polar co-ordinates of a point are `(3/4, (3pi)/4)`, then its Cartesian co-ordinate are ______
Find the polar co-ordinates of point whose Cartesian co-ordinates are `(1, sqrt(3))`
Find the Cartesian co-ordinates of point whose polar co-ordinates are `(4, pi/3)`
In ∆ABC, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a2
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 one side of a triangle is double the other and the angles opposite to these sides differ by 60°, then the triangle is ______
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 sin2A + sin2B = sin2C and l(AB) = 10, then the maximum value of the area of ΔABC is ______
In ΔABC, if `cosA/a = cosB/b,` then triangle ABC is ______
In `triangleABC,` if a = 3, b = 4, c = 5, then sin 2B = ______.
If in a `triangle"ABC",` a2cos2 A - b2 - c2 = 0, then ______.
In ΔABC, `cos"A"/"a" = cos"B"/"b" cos"C"/"c"`. If a = `1/sqrt(6)`, then the area of the triangle is ______.
In a ΔABC, if `sin"A"/sin"C" = (sin("A" - "B"))/(sin("B" - "C"))`, then a2, b2, c2 are in ______.
In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of triangle is ______.
In a ΔABC, if `("b" + "c")/11 = ("c" + "a")/12 = ("a" + "b")/13`, then cos C = ______.
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 ______.
In ΔABC with usual notations, if ∠A = 30° and a = 5, then `s/(sumsinA)` is equal to ______.
In a triangle ABC, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
In ΔABC, with usual notations, if a, b, c are in A.P. Then `a cos^2 (C/2) + c cos^2(A/2)` = ______.
The perimeter of ΔABC is 20, ∠A = 60°, area of ΔABC = `10sqrt(3)`, then find the values of a, b, 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’.
With usual notations, in a triangle ABC, if θ is any real number, then a cos(B - θ) + b cos (A + θ) is
