Advertisements
Advertisements
Question
Solve the following equations:
cot θ + cosec θ = `sqrt(3)`
Advertisements
Solution
cot θ + cosec θ = `sqrt(3)`
`cos theta/sin theta + 1/sin theta = sqrt(3)`, sin θ ≠ 0
`(cos theta + 1)/sin theta = sqrt(3)`, sin θ ≠ 0
1 + cos θ = `sqrt(3) sin theta`
`sqrt(3)sin theta - cos theta` = 1
Divide each term by 2
`sqrt(3)/2 sin theta - 1/2 cos theta = 1/2`
`sin pi/3 * sin theta - cos pi/3 * cos theta = 1/2`
`- (cos theta cos pi/3 - sin theta sin pi/3) = 1/2`
`cos (theta + pi/3) = - 1/2`
`cos (theta + pi/3) = cos (theta - pi/3)`
`cos (theta + pi/3) = cos ((3pi - pi)/3)`
`cos (theta + pi/3) = cos ((2pi)/3)`
The general solution is
`theta + pi/3 = 2"n"pi + (2pi)/3`, n ∈ Z
θ = `2"n"pi - pi/3 + (2pi)/3`, n ∈ Z
θ = `2"n"pi - pi/3 - (2pi)/3` or θ = `2"n"pi - pi/3 + (2pi)/3`
θ = `2"n"pi - (3pi)/3` or θ = `2"n"pi + (2pi - pi)/3`
θ = `2"n"pi - pi` or θ = `2"n"pi + pi/3`, n ∈ Z
θ = `(2"n" - 1)pi` or θ = `2"n"pi + pi/3`, n ∈ Z
Since sin θ ≠ 0, θ = (2n – 1)π is not possible
∴ θ = `2"n"pi + pi/3`, n ∈ Z
APPEARS IN
RELATED QUESTIONS
Find the general solution of the equation sin 2x + cos x = 0
Find the general solution of the equation sin x + sin 3x + sin 5x = 0
If \[\tan x = \frac{a}{b},\] show that
If \[\cot x \left( 1 + \sin x \right) = 4 m \text{ and }\cot x \left( 1 - \sin x \right) = 4 n,\] \[\left( m^2 + n^2 \right)^2 = mn\]
Prove the:
\[ \sqrt{\frac{1 - \sin x}{1 + \sin x}} + \sqrt{\frac{1 + \sin x}{1 - \sin x}} = - \frac{2}{\cos x},\text{ where }\frac{\pi}{2} < x < \pi\]
In a ∆ABC, prove that:
cos (A + B) + cos C = 0
Prove that:
\[\sin\frac{13\pi}{3}\sin\frac{8\pi}{3} + \cos\frac{2\pi}{3}\sin\frac{5\pi}{6} = \frac{1}{2}\]
The value of sin25° + sin210° + sin215° + ... + sin285° + sin290° 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:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
3 – 2 cos x – 4 sin x – cos 2x + sin 2x = 0
Write the number of solutions of the equation tan x + sec x = 2 cos x in the interval [0, 2π].
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:
cot θ = `sqrt(3)`
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
sin4x = sin2x
Solve the equation sin θ + sin 3θ + sin 5θ = 0
