Advertisements
Advertisements
Question
Find the general solution of the equation sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x
Advertisements
Solution
Given that: sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x
⇒ (sin3x + sinx) – 3sin2x = (cos3x + cosx) – 3cos2x
⇒ `2sin((3x + x)/2) . cos((3x - x)/2) - 3sin2x = 2cos((3x + x)/2).cos((3x - x)/2) - 3cos2x`
⇒ 2sin2x . cosx – 3sin2x = 2cos2x . cosx – 3cos2x
⇒ 2sin2x cosx – 2cos2x . cosx = 3sin2x – 3cos2x
⇒ 2cosx (sin2x – cos2x) = 3(sin2x – cos2x)
⇒ 2cosx(sin2x – cos2x) – 3(sin2x – cos2x) = 0
⇒ (sin2x – cos2x)(2cosx – 3) = 0
⇒ sin2x – cos2x = 0 and 2cosx – 3 ≠ 0 ....[∵ – 1 ≤ cos x ≤ 1]
⇒ `(sin2x)/(cos2x) - 1` = 0
⇒ tan2x = 1
⇒ tan2x = `tan pi/4`
⇒ 2x = `npi + pi/4`
∴ x = `(npi)/2 + pi/8`
Hence, the general solution of the equation is x = `(npi)/2 + pi/8`, n ∈ Z.
APPEARS IN
RELATED QUESTIONS
Find the general solution of the equation cos 3x + cos x – cos 2x = 0
Find the general solution of the equation sin x + sin 3x + sin 5x = 0
If \[\cot x \left( 1 + \sin x \right) = 4 m \text{ and }\cot x \left( 1 - \sin x \right) = 4 n,\] \[\left( m^2 + n^2 \right)^2 = mn\]
Prove that:
\[\sin^2 \frac{\pi}{18} + \sin^2 \frac{\pi}{9} + \sin^2 \frac{7\pi}{18} + \sin^2 \frac{4\pi}{9} = 2\]
In a ∆ABC, prove that:
cos (A + B) + cos C = 0
Find x from the following equations:
\[cosec\left( \frac{\pi}{2} + \theta \right) + x \cos \theta \cot\left( \frac{\pi}{2} + \theta \right) = \sin\left( \frac{\pi}{2} + \theta \right)\]
If tan \[x = - \frac{1}{\sqrt{5}}\] and θ lies in the IV quadrant, then the value of cos x is
sin2 π/18 + sin2 π/9 + sin2 7π/18 + sin2 4π/9 =
If tan A + cot A = 4, then tan4 A + cot4 A is equal to
If sec x + tan x = k, cos x =
If \[f\left( x \right) = \cos^2 x + \sec^2 x\], then
Which of the following is incorrect?
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
\[\cot x + \tan x = 2\]
Solve the following equation:
\[\sec x\cos5x + 1 = 0, 0 < x < \frac{\pi}{2}\]
Solve the following equation:
3tanx + cot x = 5 cosec x
Solve the following equation:
3 – 2 cos x – 4 sin x – cos 2x + sin 2x = 0
Write the values of x in [0, π] for which \[\sin 2x, \frac{1}{2}\]
and cos 2x are in A.P.
Write the number of points of intersection of the curves
Write the solution set of the equation
If \[2 \sin^2 x = 3\cos x\]. where \[0 \leq x \leq 2\pi\], then find the value of x.
If a is any real number, the number of roots of \[\cot x - \tan x = a\] in the first quadrant is (are).
The general solution of the equation \[7 \cos^2 x + 3 \sin^2 x = 4\] is
A solution of the equation \[\cos^2 x + \sin x + 1 = 0\], lies in the interval
The number of values of x in [0, 2π] that satisfy the equation \[\sin^2 x - \cos x = \frac{1}{4}\]
If \[e^{\sin x} - e^{- \sin x} - 4 = 0\], then x =
If \[\cos x = - \frac{1}{2}\] and 0 < x < 2\pi, then the solutions are
The number of values of x in the interval [0, 5 π] satisfying the equation \[3 \sin^2 x - 7 \sin x + 2 = 0\] is
Find the principal solution and general solution of the following:
sin θ = `-1/sqrt(2)`
Choose the correct alternative:
If tan 40° = λ, then `(tan 140^circ - tan 130^circ)/(1 + tan 140^circ * tan 130^circ)` =
Choose the correct alternative:
If sin α + cos α = b, then sin 2α is equal to
In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is ______.
