Advertisements
Advertisements
Question
Solve the following equations:
`tan theta + tan (theta + pi/3) + tan (theta + (2pi)/3) = sqrt(3)`
Advertisements
Solution
Now `tan(theta + pi/3) = (tantheta + sqrt(3))/(1 - sqrt(3) tan theta)` and `tan(theta + (2pi)/3)`
= `(tantheta - sqrt(3))/(1 + sqrt(3) tan theta)`
So, `tan(theta + pi/3) + tan(theta + (2pi)/3)`
= `(tantheta + sqrt(3))/(1 - sqrt(3) tantheta) + (tan theta - sqrt(3))/(1 + sqrt(3) tantheta)`
= `((tan theta + sqrt(3))(1 + sqrt(3) tan theta) + (tan theta - sqrt(3))(1 - sqrt(3) tan theta))/(1 - 3tan^2theta)`
= `(tan theta + sqrt(3) + sqrt(3)tan^2theta + 3tantheta + tantheta - sqrt(3)tan^2theta - sqrt(3) + 3tantheta)/(1 - 3tan^2theta)`
= `(8tantheta)/(1 - 3tan^2theta)`
Given, `tantheta + tan(theta + pi/3) + tan(theta + (2pi)/3) = sqrt(3)`
⇒ `tan theta + (8tantheta)/(1 - 3tan^2theta) = sqrt(3)`
⇒ `(tantheta - 3tan^3theta + 8tantheta)/(1 - 3tan^2theta) = sqrt(3)`
APPEARS IN
RELATED QUESTIONS
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:
Prove that
Prove that:
\[\tan 4\pi - \cos\frac{3\pi}{2} - \sin\frac{5\pi}{6}\cos\frac{2\pi}{3} = \frac{1}{4}\]
If tan x + sec x = \[\sqrt{3}\], 0 < x < π, then x is equal to
If tan A + cot A = 4, then tan4 A + cot4 A is equal to
If x sin 45° cos2 60° = \[\frac{\tan^2 60^\circ cosec30^\circ}{\sec45^\circ \cot^{2^\circ} 30^\circ}\], then x =
If \[cosec x + \cot x = \frac{11}{2}\], then tan 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:
3sin2x – 5 sin x cos x + 8 cos2 x = 2
Solve the following equation:
\[2^{\sin^2 x} + 2^{\cos^2 x} = 2\sqrt{2}\]
If secx cos5x + 1 = 0, where \[0 < x \leq \frac{\pi}{2}\], find the value of x.
Write the number of points of intersection of the curves
Write the number of values of x in [0, 2π] that satisfy the equation \[\sin x - \cos x = \frac{1}{4}\].
If \[\cot x - \tan x = \sec x\], then, x is equal to
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
Solve the equation sin θ + sin 3θ + sin 5θ = 0
Find the general solution of the equation 5cos2θ + 7sin2θ – 6 = 0
