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 principal and general solutions of the equation `tan x = sqrt3`
Find the general solution of the equation cos 4 x = cos 2 x
Find the general solution of the equation sin x + sin 3x + sin 5x = 0
If \[T_n = \sin^n x + \cos^n x\], prove that \[6 T_{10} - 15 T_8 + 10 T_6 - 1 = 0\]
Prove that
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 ∆A, B, C, D be the angles of a cyclic quadrilateral, taken in order, prove that cos(180° − A) + cos (180° + B) + cos (180° + C) − sin (90° + D) = 0
Prove that:
\[\sin\frac{13\pi}{3}\sin\frac{8\pi}{3} + \cos\frac{2\pi}{3}\sin\frac{5\pi}{6} = \frac{1}{2}\]
If \[0 < x < \frac{\pi}{2}\], and if \[\frac{y + 1}{1 - y} = \sqrt{\frac{1 + \sin x}{1 - \sin x}}\], then y is equal to
If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then x2 + y2 + z2 is independent of
If tan \[x = - \frac{1}{\sqrt{5}}\] and θ lies in the IV quadrant, then the value of cos x is
If tan A + cot A = 4, then tan4 A + cot4 A is equal to
The value of \[\tan1^\circ \tan2^\circ \tan3^\circ . . . \tan89^\circ\] is
Which of the following is correct?
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:
Find the general solution of the following equation:
Solve the following equation:
cosx + sin x = cos 2x + sin 2x
Write the number of solutions of the equation tan x + sec x = 2 cos x in the interval [0, 2π].
Write the number of solutions of the equation
\[4 \sin x - 3 \cos x = 7\]
Write the general solutions of tan2 2x = 1.
If \[2 \sin^2 x = 3\cos x\]. where \[0 \leq x \leq 2\pi\], then find the value of x.
In (0, π), the number of solutions of the equation \[\tan x + \tan 2x + \tan 3x = \tan x \tan 2x \tan 3x\] is
If \[e^{\sin x} - e^{- \sin x} - 4 = 0\], then x =
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)`
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
2 sin2x + 1 = 3 sin x
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
cos 2x = 1 − 3 sin x
Solve the following equations:
sin 5x − sin x = cos 3
Solve the following equations:
`tan theta + tan (theta + pi/3) + tan (theta + (2pi)/3) = sqrt(3)`
Choose the correct alternative:
`(cos 6x + 6 cos 4x + 15cos x + 10)/(cos 5x + 5cs 3x + 10 cos x)` is equal to
Solve 2 tan2x + sec2x = 2 for 0 ≤ x ≤ 2π.
If sin θ and cos θ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation ______.
Find the general solution of the equation 5cos2θ + 7sin2θ – 6 = 0
