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 ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a + c - b)
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
Find the Cartesian co-ordinates of the point whose polar co-ordinates are:
`(sqrt(2), pi/4)`
Find the polar coordinates of the point whose Cartesian coordinates are `(1, - sqrt(3))`.
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:
`"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 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.
If sin `(sin^-1 1/5 + cos^-1 x) = 1`, then find the value of x.
Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.
In ∆ABC, if ∠A = 30°, ∠B = 60°, then the ratio of sides is ______.
In ∆ABC, if b2 + c2 − a2 = bc, then ∠A = ______.
In ∆ABC, if sin2A + sin2B = sin2C, then show that a2 + b2 = c2
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, prove that `("b" - "c")^2 cos^2 ("A"/2) + ("b" + "c")^2 sin^2 ("A"/2)` = a2
In ∆ABC, prove that `(cos 2"A")/"a"^2 - (cos 2"c")/"c"^2 = 1/"a"^2 - 1/"c"^2`
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 ∆ABC, prove that `(cos^2"A" - cos^2"B")/("a" + "b") + (cos^2"B" - cos^2"C")/("b" + "c") + (cos^2"C" - cos^2"A")/("c" + "a")` = 0
In a ΔABC, cot `(("A - B")/2)* tan (("A + B")/2)` is equal to
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))` = ______
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 ______
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, 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 ______.
In a ΔABC, if a = `sqrt(2)` x and b = 2y and ∠C = 135°, then the area of 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 ______.
Let ABC be a triangle such that ∠A = 45°, ∠B = 75° then `"a" + "c"sqrt(2)` is equal to ______. (in usual notation)
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 = ______.
