Advertisements
Advertisements
प्रश्न
Find the general solution of the following equation:
cos 4θ = cos 2θ
Advertisements
उत्तर
The general solution of cos θ = cos α is
θ = 2nπ ± α, n ∈ Z.
∴ the general solution of cos 4θ = cos 2θ is given by
4θ = 2nπ ± 2θ, n ∈ Z
Taking positive sign, we get
4θ = 2nπ + 2θ, n ∈ Z
∴ 2θ = 2nπ, n ∈ Z
∴ θ = nπ, n ∈ Z
Taking negative sign, we get
4θ = 2nπ – 2θ, n ∈ Z
∴ 6θ = 2nπ, n ∈ Z
∴ θ = `(npi)/(3)`, n ∈ Z
Hence, the required general solution is
θ = `(npi)/(3)`,n ∈ Z or θ = nπ, n ∈ Z.
Alternative Method:
cos 4θ = cos 2θ
∴ cos 4θ – cos 2θ = 0
∴ `-2sin((4θ + 2θ)/2).sin((4θ - 2θ)/2)` = 0
∴ sin 3θ. sin θ = 0
∴ either sin 3θ = 0 or sin θ = 0
The general solution of sin θ = 0 is θ = nπ, n ∈ Z.
∴ the required general solution is given by
3θ = nπ, n ∈ Z or θ = nπ, n ∈ Z
i.e. θ = `(npi)/(3)`, n ∈ Z or θ = nπ, n ∈ Z.
APPEARS IN
संबंधित प्रश्न
Find the principal solution of the following equation:
`sqrt(3)` cosecθ + 2 = 0
Find the general solution of the following equation:
cot θ = 0.
Find the general solution of the following equation:
cosec θ = - √2.
Find the general solution of the following equation:
sin θ = tan θ
Find the general solution of the following equation:
cosθ + sinθ = 1.
State whether the following equation has a solution or not?
2sinθ = 3
State whether the following equation have solution or not?
3 tanθ = 5
Select the correct option from the given alternatives:
If polar coordinates of a point are `(2, pi/4)`, then its cartesian coordinates are
If `sqrt3 cos x − sin x = 1`, then general value of x is ______.
If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to
`"cos"^-1 ("cos" (7pi)/6)` = _________.
If `"sin"^-1 4/5 + "cos"^-1 12/13 = "sin"^-1 alpha`, then α = ______.
If tan θ + tan 2θ + tan 3θ = tan θ.tan 2θ. tan 3θ, then the general value of the θ is ______.
Find the general solutions of the following equation:
`tan^2 theta = 3`
Find the general solutions of the following equation:
sin2 θ - cos2 θ = 1
In ΔABC, prove that `("a - b")^2 cos^2 "C"/2 + ("a + b")^2 sin^2 "C"/2 = "c"^2`
In Δ ABC, if cos A = sin B - cos C then show that it is a right-angled triangle.
State whether the following equation has a solution or not?
3 sin θ = 5
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 `cot^-1 1/3 - tan^-1 1/3 = cot^-1 3/4`.
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 cosec x = 2
`int 1/(sin x * cos x)` dx = ?
If `|bar"a"|` = 10, `|bar"b"| = 2`, then `sqrt(|bar"a" xx bar"b"|^2 + |bar"a"*bar"b"|^2)` = ?
If y = sin-1 `[(sqrt(1 + x) + sqrt(1 - x))/2]`, then `"dy"/"dx"` = ?
The values of x in `(0, pi/2)` satisfying the equation sin x cos x = `1/4` are ______.
If function
f(x) = `x - |x|/x, x < 0`
= `x + |x|/x, x > 0`
= 1, x = 0, then
The measure of the angle between lines (sin2θ - 1)x2 - 2xy cos2θ + cos2θy2 = 0 is ______
If `(tan 3 theta - 1)/(tan 3 theta + 1) = sqrt3`, then the general value of θ is ______.
The general solution of 4sin2 x = 1 is ______.
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 ______.
The general solution of x(1 + y2)1/2 dx + y(1 + x2)1/2 dy = 0 is ______.
The general solution of the equation tan θ + tan 4θ + tan 7θ = tan θ tan 4θ tan 7θ is ______.
General solution of the equation sin 2x – sin 4x + sin 6x = 0 is ______.
If a = sin θ + cos θ, b = sin3 θ + cos3 θ, then ______.
If `tanx/(tan 2x) + (tan 2x)/tanx + 2` = 0, then the general value of x is ______.
