Advertisements
Advertisements
Question
Solve the following equations:
cot θ + cosec θ = `sqrt(3)`
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 principal and general solutions of the equation `cot x = -sqrt3`
If \[cosec x - \sin x = a^3 , \sec x - \cos x = b^3\], then prove that \[a^2 b^2 \left( a^2 + b^2 \right) = 1\]
Prove that: cos 24° + cos 55° + cos 125° + cos 204° + cos 300° = \[\frac{1}{2}\]
In a ∆ABC, prove that:
Prove that:
If tan \[x = - \frac{1}{\sqrt{5}}\] and θ lies in the IV quadrant, then the value of cos x is
If \[cosec x + \cot x = \frac{11}{2}\], then tan x =
If \[f\left( x \right) = \cos^2 x + \sec^2 x\], then
Which of the following is incorrect?
Which of the following is correct?
Find the general solution of the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
\[\sec x\cos5x + 1 = 0, 0 < x < \frac{\pi}{2}\]
Solve the following equation:
sin x tan x – 1 = tan x – sin x
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
If \[\cos x = - \frac{1}{2}\] and 0 < x < 2\pi, then the solutions are
Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0
Choose the correct alternative:
If cos pθ + cos qθ = 0 and if p ≠ q, then θ is equal to (n is any integer)
