Advertisements
Advertisements
Question
If cos-1 x + cos-1y + cos-1z = 3π, then show that x2 + y2 + z2 + 2xyz = 1.
Advertisements
Solution
0 ≤ cos-1x ≤ π and
cos-1x + cos-1 y + cos-1z = 3π
∴ cos-1x = π, cos-1y = π and cos-1z = π
∴ x = y = z = cos π = - 1
∴ x2 + y2 + z2 + 2xyz
= (- 1)2 + (- 1)2 + (- 1)2 + 2(- 1)(- 1)(- 1)
= 1 + 1 + 1 - 2
= 3 - 2
= 1.
APPEARS IN
RELATED QUESTIONS
Find the principal solution of the following equation:
tan θ = – 1
Find the principal solution of the following equation:
`sqrt(3)` cosecθ + 2 = 0
Find the general solution of the following equation :
cosθ = `sqrt(3)/(2)`
Find the general solution of the following equation:
tan θ = `(1)/(sqrt(3))`
Find the general solution of the following equation:
tan `(2θ)/(3) = sqrt3`
Find the general solution of the following equation:
cot 4θ = – 1
Find the general solution of the following equation:
cosθ + sinθ = 1.
State whether the following equation has a solution or not?
cos2θ = – 1.
State whether the following equation has a solution or not?
2sinθ = 3
In Δ ABC, prove that a3 sin(B – C) + b3sin(C – A) + c3sin(A – B) = 0
With usual notations prove that 2(bc cosA + ac cosB + ab cosC) = a2 + b2 + c2 .
Select the correct option from the given alternatives:
The principal solutions of equation sin θ = `- 1/2` are ______.
Select the correct option from the given alternatives:
In ΔABC if c2 + a2 – b2 = ac, then ∠B = ____.
Select the correct option from the given alternatives:
In ΔABC, ac cos B - bc cos A = _______
Select the correct option from the given alternatives:
`2 "tan"^-1 (1/3) + "tan"^-1 (1/7) =` _____
Find the principal solutions of the following equation:
sin 2θ = `-1/2`
Find the principal solutions of the following equation:
cot θ = 0
In Δ ABC, prove that `cos(("A" - "B")/2) = (("a" + "b")/"c")sin "C"/2` .
In Δ ABC, if cos A = sin B - cos C then show that it is a right-angled triangle.
If 2 tan-1(cos x) = tan-1(2 cosec x), then find the value of x.
If sin-1(1 - x) - 2 sin-1x = `pi/2`, then find the value of x.
If tan-12x + tan-13x = `pi/4`, then find the value of x.
Show that `tan^-1 1/2 = 1/3 tan^-1 11/2`
Show that `2 cot^(-1) 3/2 + sec^(-1) 13/12 = π/2`
Prove the following:
`cos^-1 "x" = tan^-1 (sqrt(1 - "x"^2)/"x")`, if x > 0
If | x | < 1, then prove that
`2 tan^-1 "x" = tan^-1 ("2x"/(1 - "x"^2)) = sin^-1 ("2x"/(1 + "x"^2)) = cos^-1 ((1 - "x"^2)/(1 + "x"^2))`
If x, y, z are positive, then prove that
`tan^-1 (("x - y")/(1 + "xy")) + tan^-1 (("y - z")/(1 + "yz")) + tan^-1 (("z - x")/(1 + "zx")) = 0`
Find the principal solutions of cos 2𝑥 = 1
Find the principal solutions of tan x = `-sqrt(3)`
If cos–1x + cos–1y – cos–1z = 0, then show that x2 + y2 + z2 – 2xyz = 1
`cos^-1 (cos (4pi)/3)` = ______.
`[sin (tan^-1 3/4)]^2 + [sin(tan^-1 4/3)]^2 = ?`
If y = sin-1 `[(sqrt(1 + x) + sqrt(1 - x))/2]`, then `"dy"/"dx"` = ?
The number of solutions of cos 2θ = sin θ in (0, 2π) is ______
The general solution of sin 2x = cos 2x is ______
The general solution of the equation tan θ tan 2θ = 1 is given by ______
Which of the following is true in a triangle ABC?
The general solution of cosec x = `-sqrt2` is ______
The equation 3sin2x + 10 cos x – 6 = 0 is satisfied, if ______.
The number of solutions of sin x + sin 3x + sin 5x = 0 in the interval `[pi/2, (3pi)/2]` is ______.
If 2 tan–1(cos x) = tan–1(2 cosec x). then find the value of x.
The general solution of x(1 + y2)1/2 dx + y(1 + x2)1/2 dy = 0 is ______.
Which of the following equation has no solution?
If a = sin θ + cos θ, b = sin3 θ + cos3 θ, then ______.
If `sin^-1 4/5 + cos^-1 12/13` = sin–1α, then find the value of α.
