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
Prove that: tan 225° cot 405° + tan 765° cot 675° = 0
Prove that
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
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 =
The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° is
If x sin 45° cos2 60° = \[\frac{\tan^2 60^\circ cosec30^\circ}{\sec45^\circ \cot^{2^\circ} 30^\circ}\], then 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:
Solve the following equation:
Solve the following equation:
Solve the following equation:
`cosec x = 1 + cot x`
Write the number of points of intersection of the curves
If a is any real number, the number of roots of \[\cot x - \tan x = a\] in the first quadrant is (are).
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
Solve the following equations:
cos 2θ = `(sqrt(5) + 1)/4`
If a cosθ + b sinθ = m and a sinθ - b cosθ = n, then show that a2 + b2 = m2 + n2
