Advertisements
Advertisements
प्रश्न
Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0
Advertisements
उत्तर
sin θ + sin 3θ + sin 5θ = 0
`2 sin ((5theta + theta)/2) * cos ((5theta - theta)/2) + sin 3theta` = 0
`2sin ((6theta)/2) * cos ((4theta)/2) + sin 3theta` = 0
2 sin 3θ . cos 2θ + sin 3θ = 0
sin 3θ (2 cos 2θ + 1) = θ
sin 3θ = 0 or 2 cos 2θ + 1 = θ
sin 3θ = 0 or cos 2θ = `- 1/2`
To find the general solution of sin 3θ = 0
The general solution is
3θ = nπ, n ∈ Z
θ = `("n"pi)/3`, n ∈ Z
To find the general solution of cos 2θ = ` - 1/2`
cos 2θ = ` - 1/2`
cos 2θ = `cos (pi - pi/3)`
cos 2θ = `cos ((3pi - pi)/3)`
cos 2θ = `cos ((2pi)/3)`
The general solution is
2θ = `2"n"pi +- (2pi)/3`, n ∈ Z
θ = `"n"pi +- pi/3`, n ∈ Z
∴ The required solutions are
θ = `(:"n"pi)/3`, n ∈ Z
or
θ = `"n"pi +- pi/3`, n ∈ Z
APPEARS IN
संबंधित प्रश्न
Prove that
In a ∆ABC, prove that:
Find x from the following equations:
\[cosec\left( \frac{\pi}{2} + \theta \right) + x \cos \theta \cot\left( \frac{\pi}{2} + \theta \right) = \sin\left( \frac{\pi}{2} + \theta \right)\]
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 \[\frac{3\pi}{4} < \alpha < \pi, \text{ then }\sqrt{2\cot \alpha + \frac{1}{\sin^2 \alpha}}\] is equal to
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
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 =
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:
Solve the following equation:
sin x tan x – 1 = tan x – sin x
Solve the following equation:
\[2^{\sin^2 x} + 2^{\cos^2 x} = 2\sqrt{2}\]
The general value of x satisfying the equation
\[\sqrt{3} \sin x + \cos x = \sqrt{3}\]
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
2 cos2x + 1 = – 3 cos x
Solve the following equations:
sin θ + cos θ = `sqrt(2)`
Solve `sqrt(3)` cos θ + sin θ = `sqrt(2)`
Find the general solution of the equation sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x
