Advertisements
Advertisements
Question
Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0
Advertisements
Solution
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
RELATED QUESTIONS
Find the principal and general solutions of the equation sec x = 2
If \[\cot x \left( 1 + \sin x \right) = 4 m \text{ and }\cot x \left( 1 - \sin x \right) = 4 n,\] \[\left( m^2 + n^2 \right)^2 = mn\]
If \[T_n = \sin^n x + \cos^n x\], prove that \[\frac{T_3 - T_5}{T_1} = \frac{T_5 - T_7}{T_3}\]
Prove that
In a ∆ABC, prove that:
cos (A + B) + cos C = 0
In a ∆ABC, prove that:
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:
Solve the following equation:
\[2 \sin^2 x = 3\cos x, 0 \leq x \leq 2\pi\]
Write the number of points of intersection of the curves
If \[2 \sin^2 x = 3\cos x\]. where \[0 \leq x \leq 2\pi\], then find the value of x.
A solution of the equation \[\cos^2 x + \sin x + 1 = 0\], lies in the interval
In (0, π), the number of solutions of the equation \[\tan x + \tan 2x + \tan 3x = \tan x \tan 2x \tan 3x\] is
Find the principal solution and general solution of the following:
sin θ = `-1/sqrt(2)`
Solve the equation sin θ + sin 3θ + sin 5θ = 0
