Advertisements
Advertisements
Question
If sin `(sin^-1 1/5 + cos^-1 x) = 1`, then find the value of x.
Solution
sin `(sin^-1 1/5 + cos^-1 x) = 1`
∴ `sin^-1 1/5 + cos^-1 "x" = sin^-1 (1)`
∴ `sin^-1 1/5 + cos^-1 "x" = sin^-1 (sin pi / 2)`
∴ `sin^-1 1/5 + cos^-1 "x" = pi/2`
∴ x = `1/5 .....[because sin^-1"x" + cos^-1 "x" = pi/2]`
APPEARS IN
RELATED QUESTIONS
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 a Δ ABC, with usual notations prove that:` (a -bcos C) /(b -a cos C )= cos B/ cos A`
In ΔABC, prove that `tan((A - B)/2) = (a - b)/(a + b)*cot C/2`
In ΔABC with usual notations, prove that 2a `{sin^2(C/2)+csin^2 (A/2)}` = (a + c - b)
If in ∆ABC with usual notations a = 18, b = 24, c = 30 then sin A/2 is equal to
(A) `1/sqrt5`
(B) `1/sqrt10`
(C) `1/sqrt15`
(D) `1/(2sqrt5)`
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 ,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:
`(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.
`(0, 1/2)`
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 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 Δ ABC, if sin2 A + sin2 B = sin2 C, then show that the triangle is a right-angled triangle.
Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]
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`
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 ∠A = 30°, ∠B = 60°, then the ratio of sides is ______.
In ∆ABC, if b2 + c2 − a2 = bc, then ∠A = ______.
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, if `(2cos "A")/"a" + (cos "B")/"b" + (2cos"C")/"c" = "a"/"bc" + "b"/"ca"`, then show that the triangle is a right angled
In ΔABC, a(cos2B + cos2C) + cos A(c cos C + b cos B) = ?
In a ΔABC if 2 cos C = sin B · cosec A, then ______.
In a triangle ABC with usual notations, if `(cos "A")/"a" = (cos "B")/"b" = (cos "C")/"c"`, then area of triangle ABC with a = `sqrt6` is ____________.
In Δ ABC; with usual notations, if cos A = `(sin "B")/(sin "C")`, then the triangle is _______.
In a ΔABC, 2ab sin`((A + B - C)/2)` = ______
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 usual notations, `("b" sin "B" - "c" sin "C")/(sin ("B - C"))` = _______.
In ΔABC, if `cosA/a = cosB/b,` then triangle ABC is ______
The smallest angle of the ΔABC, when a = 7, b = `4sqrt(3)` and c = `sqrt(13)` 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 ______.
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 ______
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 ______.
Find the cartesian co-ordinates of the point whose polar co-ordinates are `(1/2, π/3)`.
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 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)
In a triangle ABC, in usual notation, (a + b + c)(b + c – a) = λbc will be true if ______.
In a triangle ABC, ∠C = 90°, then `(a^2 - b^2)/(a^2 + b^2)` is ______.
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’.
