Advertisements
Advertisements
Question
Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0
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 `cot x = -sqrt3`
Find the general solution of the equation cos 4 x = cos 2 x
If \[\tan x = \frac{b}{a}\] , then find the values of \[\sqrt{\frac{a + b}{a - b}} + \sqrt{\frac{a - b}{a + b}}\].
If \[\tan x = \frac{a}{b},\] show that
If \[cosec x - \sin x = a^3 , \sec x - \cos x = b^3\], then prove that \[a^2 b^2 \left( a^2 + b^2 \right) = 1\]
If \[a = \sec x - \tan x \text{ and }b = cosec x + \cot x\], then shown that \[ab + a - b + 1 = 0\]
Prove the:
\[ \sqrt{\frac{1 - \sin x}{1 + \sin x}} + \sqrt{\frac{1 + \sin x}{1 - \sin x}} = - \frac{2}{\cos x},\text{ where }\frac{\pi}{2} < x < \pi\]
Prove that:
\[\frac{\cos (2\pi + x) cosec (2\pi + x) \tan (\pi/2 + x)}{\sec(\pi/2 + x)\cos x \cot(\pi + x)} = 1\]
Prove that:
If tan x = \[x - \frac{1}{4x}\], then sec x − tan x is equal to
The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° is
Find the general solution of the following equation:
Write the number of solutions of the equation
\[4 \sin x - 3 \cos x = 7\]
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
2 cos2x + 1 = – 3 cos x
Choose the correct alternative:
If cos pθ + cos qθ = 0 and if p ≠ q, then θ is equal to (n is any integer)
Choose the correct alternative:
If tan α and tan β are the roots of x2 + ax + b = 0 then `(sin(alpha + beta))/(sin alpha sin beta)` is equal to
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.
