Advertisements
Advertisements
प्रश्न
Solve the following equations:
sin 2θ – cos 2θ – sin θ + cos θ = θ
Advertisements
उत्तर
sin 2θ – cos 2θ – sin θ + cos θ = θ
`2cos ((2theta + theta)/2) sin ((2theta - theta)/2) - 2 sin ((2theta + theta)/2) sin ((theta - 2theta)/2)` = 0
`2cos ((3theta)/2) * sin (theta/2) - 2sin ((3theta)/2) sin (- theta/2)` = 0
`2cos ((3theta)/2) * sin (theta/2) + 2sin ((3theta)/2) sin (theta/2)` = 0
`2sin theta/2 [cos ((3theta)/2) + sin ((3theta)/2)]` = 0
`2 sin theta/2` = 0 or `cos ((3theta)/2) + sin ((3theta)/2)` = 0
`sin theta/2` = 0 or `cos ((3theta)/2) = - sin ((3theta)/2)`
`sin theta/2` = 0 or `(sin ((3theta)/2))/(cos ((3theta)/2))` = – 1
`sin theta/2` = 0 or `tan ((3theta)/2)` = – 1
To find the general solution of `sin theta/2` = 0
The general solution is
`theta/2` = nπ, n ∈ Z
θ = 2nπ, n ∈ Z
To find the general solution of `tan ((3theta)/2)` = – 1
`tan ((3theta)/2)` = – 1
`tan ((3theta)/2) = tan (pi - pi/4)`
`tan ((3theta)/2) = tan ((4pi - pi)/4)`
`tan ((3theta)/2) = tan ((3pi)/4)`
The general solution is
`(3theta)/2 = "n" + pi/4`, n ∈ Z
θ = `(2"n"pi)/3 + (2pi)/(3 xx 4)`, n ∈ Z
θ = `(2"n"pi)/3 + pi/6`, n ∈ Z
∴ The required solutions are
θ = 2nπ, n ∈ Z
or
θ = `(2"n"pi)/3 + pi/6`, n ∈ Z
APPEARS IN
संबंधित प्रश्न
Find the principal and general solutions of the equation `cot x = -sqrt3`
If \[T_n = \sin^n x + \cos^n x\], prove that \[2 T_6 - 3 T_4 + 1 = 0\]
Prove that: tan (−225°) cot (−405°) −tan (−765°) cot (675°) = 0
Prove that
Prove that:
\[\sin \frac{13\pi}{3}\sin\frac{2\pi}{3} + \cos\frac{4\pi}{3}\sin\frac{13\pi}{6} = \frac{1}{2}\]
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 =
Find the general solution of the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
\[2 \sin^2 x = 3\cos x, 0 \leq x \leq 2\pi\]
Solve the following equation:
\[2^{\sin^2 x} + 2^{\cos^2 x} = 2\sqrt{2}\]
Write the number of solutions of the equation
\[4 \sin x - 3 \cos x = 7\]
If cos x = k has exactly one solution in [0, 2π], then write the values(s) of k.
The general value of x satisfying the equation
\[\sqrt{3} \sin x + \cos x = \sqrt{3}\]
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
sin4x = sin2x
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
cos 2x = 1 − 3 sin x
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 ______.
