Advertisements
Advertisements
Question
Find the general solution of the following equation:
tan3θ = 3 tanθ.
Advertisements
Solution
tan3θ = 3tanθ
∴ tan3θ - 3tanθ = 0
∴ tanθ (tan2θ - 3) = 0
∴ either tanθ = 0 or tan2θ - 3 = 0
∴ either tanθ = 0 or tan2θ = 3
∴ either tanθ = 0 or tan2θ = (√3)2
∴ either tan θ = 0 or tan2θ = `(tan pi/3)^2 ...[∵ tan pi/(3) = sqrt(3)]`
∴ either tanθ = 0 or tan2θ = `tan^2 pi/(3)`
The general solution of
tanθ = 0 is θ = nπ, n ∈ Z and
tan2θ = tan2α is θ = nπ ± α, n ∈ Z.
∴ the required general solution is given by
θ = nπ, n ∈ Z or θ = `nπ ± pi/(3)`, n ∈ Z.
APPEARS IN
RELATED QUESTIONS
Find the principal solution of the following equation:
cot θ = 0
Find the principal solution of the following equation:
sin θ = `-1/2`
Find the general solution of the following equation:
sinθ = `1/2`.
Find the general solution of the following equation:
sin 2θ = `1/2`
Find the general solution of the following equation:
cot 4θ = – 1
Find the general solution of the following equation:
4 cos2 θ = 3
State whether the following equation has a solution or not?
2sinθ = 3
In ΔABC, if a cos A = b cos B then prove that the triangle is either a right angled or an isosceles traingle.
With usual notations prove that 2(bc cosA + ac cosB + ab cosC) = a2 + b2 + c2 .
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:
`"tan"(2"tan"^-1 (1/5) - pi/4)` = ______
Select the correct option from the given alternatives:
In any ΔABC, if acos B = bcos A, then the triangle is
Find the principal solutions of the following equation:
cot θ = 0
Find the general solutions of the following equation:
sin θ - cos θ = 1
In Δ ABC, prove that `cos(("A" - "B")/2) = (("a" + "b")/"c")sin "C"/2` .
If `(sin "A")/(sin "C") = (sin ("A - B"))/(sin ("B - C"))`, then show that a2, b2, c2 are in A.P.
Prove the following:
`cos^-1 "x" = pi + tan^-1 (sqrt(1 - "x"^2)/"x")`, if x < 0
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`
If `tan^-1 "x" + tan^-1 "y" + tan^-1 "z" = pi/2,` then show that xy + yz + zx = 1
The principal solutions of `sqrt(3)` sec x − 2 = 0 are ______
Find the principal solutions of sin x − 1 = 0
If cos–1x + cos–1y – cos–1z = 0, then show that x2 + y2 + z2 – 2xyz = 1
`int 1/(sin x * cos x)` dx = ?
The general solution of sec θ = `sqrt2` is
`int (sin (log x))^2/x` log x dx = ?
`[sin (tan^-1 3/4)]^2 + [sin(tan^-1 4/3)]^2 = ?`
If x + y = `pi/2`, then the maximum value of sin x. sin y is.
The principal solutions of cot x = `sqrt3` are ______.
The values of x in `(0, pi/2)` satisfying the equation sin x cos x = `1/4` are ______.
The number of solutions of `sin^2 theta = 1/2` in [0, π] is ______.
Which of the following equations has no solution?
The value of `tan^-1 1/3 + tan^-1 1/5 + tan^-1 1/7 + tan^-1 1/8` is ______.
The number of solutions of cos 2θ = sin θ in (0, 2π) is ______
If function
f(x) = `x - |x|/x, x < 0`
= `x + |x|/x, x > 0`
= 1, x = 0, then
The value of sin 18° is ______.
The general solution of 4sin2 x = 1 is ______.
The general value of θ in the equation `2sqrt(3) cos theta = tan theta` is ______.
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.
If y = `(2sinα)/(1 + cosα + sinα)`, then value of `(1 - cos α + sin α)/(1 + sin α)` is ______.
General solution of the equation sin 2x – sin 4x + sin 6x = 0 is ______.
Principal solutions at the equation sin 2x + cos 2x = 0, where π < x < 2 π are ______.
If a = sin θ + cos θ, b = sin3 θ + cos3 θ, then ______.
The general solution to cos100x – sin100x = 1 is ______.
