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 general solution for each of the following equations sec2 2x = 1– tan 2x
Find the general solution of the equation sin x + sin 3x + sin 5x = 0
Prove that: tan 225° cot 405° + tan 765° cot 675° = 0
In a ∆ABC, prove that:
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 sec \[x = x + \frac{1}{4x}\], then sec x + tan x =
If x is an acute angle and \[\tan x = \frac{1}{\sqrt{7}}\], then the value of \[\frac{{cosec}^2 x - \sec^2 x}{{cosec}^2 x + \sec^2 x}\] is
If A lies in second quadrant 3tan A + 4 = 0, then the value of 2cot A − 5cosA + sin A is equal to
Find the general solution of the following equation:
Find the general solution of the following equation:
Solve the following equation:
\[\sin^2 x - \cos x = \frac{1}{4}\]
Solve the following equation:
\[\cot x + \tan x = 2\]
Write the general solutions of tan2 2x = 1.
General solution of \[\tan 5 x = \cot 2 x\] is
Find the principal solution and general solution of the following:
sin θ = `-1/sqrt(2)`
Solve the following equations:
sin 5x − sin x = cos 3
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
Find the general solution of the equation 5cos2θ + 7sin2θ – 6 = 0
