Advertisements
Advertisements
प्रश्न
Solve the following equation:
Advertisements
उत्तर
\[ \cos x \cos2x \cos3x = \frac{1}{4}\]
\[ \Rightarrow \left[ \frac{\cos\left( x + 2x \right) + \cos\left( 2x - x \right)}{2} \right]\cos3x = \frac{1}{4}\]
\[ \Rightarrow 2\left[ \cos3x + \cos x \right]\cos3x = 1\]
\[ \Rightarrow 2 \cos^2 3x + 2\cos x \cos3x - 1 = 0\]
\[ \Rightarrow 2 \cos^2 3x - 1 + 2\cos x \cos3x = 0\]
\[ \Rightarrow \cos6x + \cos4x + \cos2x = 0\]
\[ \Rightarrow \cos6x + \cos2x + \cos4x = 0\]
\[ \Rightarrow 2\cos4xcos2x + \cos4x = 0\]
\[ \Rightarrow \cos4x\left( 2\cos2x + 1 \right) = 0\]
\[ \Rightarrow \cos4x = 0 or 2\cos2x + 1 = 0\]
\[ \Rightarrow \cos4x = 0 or \cos2x = \frac{- 1}{2}\]
\[ \Rightarrow \cos4x = \cos\frac{\pi}{2} or \cos2x = \cos\frac{2\pi}{3}\]
\[ \Rightarrow 4x = \left( 2n + 1 \right)\frac{\pi}{2}, n \in Z or 2x = 2m\pi \pm \frac{2\pi}{3}, m \in Z\]
\[ \Rightarrow x = \left( 2n + 1 \right)\frac{\pi}{8}, n \in Z or x = m\pi \pm \frac{\pi}{3}, m \in Z\]
APPEARS IN
संबंधित प्रश्न
Find the general solution of the equation cos 4 x = cos 2 x
If \[cosec x - \sin x = a^3 , \sec x - \cos x = b^3\], then prove that \[a^2 b^2 \left( a^2 + b^2 \right) = 1\]
If \[\sin x + \cos x = m\], then prove that \[\sin^6 x + \cos^6 x = \frac{4 - 3 \left( m^2 - 1 \right)^2}{4}\], where \[m^2 \leq 2\]
If \[a = \sec x - \tan x \text{ and }b = cosec x + \cot x\], then shown that \[ab + a - b + 1 = 0\]
If \[T_n = \sin^n x + \cos^n x\], prove that \[2 T_6 - 3 T_4 + 1 = 0\]
Prove that:
Prove that: tan (−225°) cot (−405°) −tan (−765°) cot (675°) = 0
Prove that
Prove that
Find x from the following equations:
\[x \cot\left( \frac{\pi}{2} + \theta \right) + \tan\left( \frac{\pi}{2} + \theta \right)\sin \theta + cosec\left( \frac{\pi}{2} + \theta \right) = 0\]
If sec \[x = x + \frac{1}{4x}\], then sec x + tan x =
If \[\frac{3\pi}{4} < \alpha < \pi, \text{ then }\sqrt{2\cot \alpha + \frac{1}{\sin^2 \alpha}}\] is equal to
If tan A + cot A = 4, then tan4 A + cot4 A is equal to
If x sin 45° cos2 60° = \[\frac{\tan^2 60^\circ cosec30^\circ}{\sec45^\circ \cot^{2^\circ} 30^\circ}\], then x =
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:
Solve the following equation:
Solve the following equation:
Solve the following equation:
sin x tan x – 1 = tan x – sin x
Write the general solutions of tan2 2x = 1.
The number of solution in [0, π/2] of the equation \[\cos 3x \tan 5x = \sin 7x\] is
If \[e^{\sin x} - e^{- \sin x} - 4 = 0\], then x =
The equation \[3 \cos x + 4 \sin x = 6\] has .... solution.
If \[\sqrt{3} \cos x + \sin x = \sqrt{2}\] , then general value of x is
If \[\cos x = - \frac{1}{2}\] and 0 < x < 2\pi, then the solutions are
Find the principal solution and general solution of the following:
tan θ = `- 1/sqrt(3)`
Solve the following equations:
cos θ + cos 3θ = 2 cos 2θ
Choose the correct alternative:
`(cos 6x + 6 cos 4x + 15cos x + 10)/(cos 5x + 5cs 3x + 10 cos x)` is equal to
If a cosθ + b sinθ = m and a sinθ - b cosθ = n, then show that a2 + b2 = m2 + n2
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.
Find the general solution of the equation sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x
In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is ______.
