Advertisements
Advertisements
प्रश्न
Show that `2 sin^-1 (3/5) = tan^-1(24/7)`
Advertisements
उत्तर
Let `2 sin^-1 (3/5)` = x
Then sin x = `3/5`, where `0 < "x" < pi/2`
∴ cos x > 0
Now, cos x = `sqrt(1 - sin^2"x") = sqrt(1 - 9/25) = sqrt(16/25) = 4/5`
∴ `tan "x" = "sin x"/"cos x" = (3/5)/(4/5) = 3/4`
∴ x = `tan^-1(3/4)`
∴ `sin^-1 (3/5) = tan^-1(3/4)`
Now, LHS = `2sin^-1 (3/5) = 2tan^-1(3/4)`
`= tan^-1 (3/4) + tan^-1(3/4)`
= `tan^-1 [(3/4 + 3/4)/(1 - 3/4 xx 3/4)] = tan^-1 [(12 + 12)/(16 - 9)]`
`= tan^-1(24/7)` = RHS
Alternative Method:
LHS = `2sin^-1 (3/5) = 2tan^-1(3/4)`
`= tan^-1 [(2(3/4))/(1 - (3/4)^2)] .....[because 2 tan^-1 "x" = tan^-1 ("2x"/(1 - "x"^2))]`
`= tan^-1 [(3/2)/(1 - (9/16))]`
`= tan^-1 (3/2 xx 16/7)`
`= tan^-1 (24/7)`
= RHS
APPEARS IN
संबंधित प्रश्न
In Δ ABC with the usual notations prove that `(a-b)^2 cos^2(C/2)+(a+b)^2sin^2(C/2)=c^2`
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 Δ ABC, if a = 13, b = 14 and c = 15, then sin (A/2)= _______.
(A) `1/5`
(B) `sqrt(1/5)`
(C) `4/5`
(D) `2/5`
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 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)`
In ΔABC, if cot A, cot B, cot C are in A.P. then show that a2, b2, c2 are also in A.P.
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 ΔABC, if `"cos A"/"a" = "cos B"/"b"`, then show that it is an isosceles triangle.
With the usual notations, show that
(c2 − a2 + b2) tan A = (a2 − b2 + c2) tan B = (b2 − c2 + a2) tan C
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.
If `tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4`, find the value of x.
In ∆ABC, if ∠A = 30°, ∠B = 60°, then the ratio of sides is ______.
In ∆ABC, if b2 + c2 − a2 = bc, then ∠A = ______.
In ∆ABC, prove that ac cos B − bc cos A = a2 − b2
In ∆ABC, if sin2A + sin2B = sin2C, then show that a2 + b2 = c2
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 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 ΔABC if 2 cos C = sin B · cosec A, then ______.
In Δ ABC; with usual notations, `("b" sin "B" - "c" sin "C")/(sin ("B - C"))` = _______.
The smallest angle of the ΔABC, when a = 7, b = `4sqrt(3)` and c = `sqrt(13)` is ______.
If in Δ ABC, 3a = b + c, then `cot ("B"/2) cot ("C"/2)` = ______.
In ΔABC, if `"a" cos^2 "C"/2 + "c" cos^2 "A"/2 = (3"b")/2`, then a, b, c are in ______.
In ΔABC, `cos"A"/"a" = cos"B"/"b" cos"C"/"c"`. If a = `1/sqrt(6)`, then the area of the triangle is ______.
If a = 13, b = 14, c = 15, then `cos("A"/2)` = ______.
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 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 a triangle ABC, with usual notations, if \[\frac{2\cos\mathrm{A}}{\mathrm{a}}+\frac{\cos\mathrm{B}}{\mathrm{b}}+\frac{2\cos\mathrm{C}}{\mathrm{c}}=\frac{\mathrm{a}}{\mathrm{bc}}+\frac{\mathrm{b}}{\mathrm{ac}}\]then ∠A =
In a triangle ABC with usual notations, if a,b, and c are in arithmetic progression, then, \[\tan\frac{A}{2}\cdot\tan\frac{C}{2}=\]
