Advertisements
Advertisements
Question
If `sqrt3 cos x − sin x = 1`, then general value of x is ______.
Options
`2npi +- pi/3`
`2npi +- pi/6`
`2npi +- pi/3 - pi/6`
`npi + (- 1)^n pi/3`
Advertisements
Solution
If `sqrt3cos x − sin x = 1`, then general value of x is `bbunderline(2npi +- pi/3 - pi/6)`.
Explanation:
Divide the entire equation `sqrt3cos x − sin x = 1` by 2,
`sqrt3/2 cos x − 1/2 sin x = 1/2`
Substitute `cos (pi/6) = sqrt3/2 and sin (pi/6) = 1/2`
`cosx cos(pi/6) - sinx sin(pi/6) = 1/2`
Using the identity cos(A + B) = cosA cosB − sinA sin B
`cos(x + pi/6) = 1/2`
The general solution for cos θ = cos α is θ = 2nπ ± α. Since `cos(pi/3) = 1/2`
`x + pi/6 = 2npi ± pi/3`
Subtract `pi/6` from both sides to get your exact statement:
`x = 2npi ± pi/3 - pi/6`
APPEARS IN
RELATED QUESTIONS
Find the principal solution of the following equation:
sin θ = `-1/2`
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:
tan θ = `(1)/(sqrt(3))`
Find the general solution of the following equation:
sec θ = `sqrt(2)`.
Find the general solution of the following equation:
tan θ = - 1
Find the general solution of the following equation:
sin 2θ = `1/2`
Find the general solution of the following equation:
4sin2θ = 1.
Find the general solution of the following equation:
tan3θ = 3 tanθ.
Find the general solution of the following equation:
cosθ + sinθ = 1.
State whether the following equation have solution or not?
cos 2θ = – 1
With the usual notations prove that `2{asin^2 "C"/(2) + "c"sin^2 "A"/(2)}` = a – b + c.
In Δ ABC, prove that a3 sin(B – C) + b3sin(C – A) + c3sin(A – B) = 0
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:
If polar coordinates of a point are `(2, pi/4)`, then its cartesian coordinates are
If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.
Select the correct option from the given alternatives:
If tan-1(2x) + tan-1(3x) = `pi/4`, then x = _____
Find the principal solutions of the following equation:
tan 3θ = - 1
Find the general solutions of the following equation:
`tan theta = - sqrt3`
Find the general solutions of the following equation:
sin2 θ - cos2 θ = 1
If tan-12x + tan-13x = `pi/4`, then find the value of x.
Show that `cot^-1 1/3 - tan^-1 1/3 = cot^-1 3/4`.
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`
If cos-1 x + cos-1y + cos-1z = 3π, then show that x2 + y2 + z2 + 2xyz = 1.
Find the principal solutions of tan x = `-sqrt(3)`
If sin-1 x = `pi/10`, for some x ∈ [-1, 1], then the value of cos-1 x is _______.
If f(x) = sin-1`(sqrt((1 - x)/2))`, then f'(x) = ?
If y = sin-1 `[(sqrt(1 + x) + sqrt(1 - x))/2]`, then `"dy"/"dx"` = ?
The number of solutions of `sin^2 theta = 1/2` in [0, π] is ______.
The value of `tan^-1 1/3 + tan^-1 1/5 + tan^-1 1/7 + tan^-1 1/8` is ______.
If 4 sin-1x + 6 cos-1 x = 3π then x = ______.
The general solution of the equation tan θ tan 2θ = 1 is given by ______
The general solution of cot θ + tan θ = 2 is ______.
If `(tan 3 theta - 1)/(tan 3 theta + 1) = sqrt3`, then the general value of θ is ______.
The general solution of cosec x = `-sqrt2` is ______
Find the principal solutions of cot θ = 0
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?
Principal solutions at the equation sin 2x + cos 2x = 0, where π < x < 2 π are ______.
Find the general solution of sin θ + sin 3θ + sin 5θ = 0
Prove that the general solution of cos θ = cos α is θ = 2nπ ± α, n ∈ Z.
If `tanx/(tan 2x) + (tan 2x)/tanx + 2` = 0, then the general value of x is ______.
If `sin^-1 4/5 + cos^-1 12/13` = sin–1α, then find the value of α.
