Advertisements
Advertisements
प्रश्न
Solve the following equations:
`tan theta + tan (theta + pi/3) + tan (theta + (2pi)/3) = sqrt(3)`
Advertisements
उत्तर
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
संबंधित प्रश्न
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:
\[\tan 4\pi - \cos\frac{3\pi}{2} - \sin\frac{5\pi}{6}\cos\frac{2\pi}{3} = \frac{1}{4}\]
Prove that:
If \[\frac{\pi}{2} < x < \pi, \text{ then }\sqrt{\frac{1 - \sin x}{1 + \sin x}} + \sqrt{\frac{1 + \sin x}{1 - \sin x}}\] is equal to
If \[cosec x + \cot x = \frac{11}{2}\], then tan x =
Find the general solution of 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 solutions of the equation
\[4 \sin x - 3 \cos x = 7\]
Write the solution set of the equation
If \[3\tan\left( x - 15^\circ \right) = \tan\left( x + 15^\circ \right)\] \[0 < x < 90^\circ\], find θ.
The smallest value of x satisfying the equation
The equation \[3 \cos x + 4 \sin x = 6\] has .... solution.
The number of values of x in the interval [0, 5 π] satisfying the equation \[3 \sin^2 x - 7 \sin x + 2 = 0\] is
Find the principal solution and general solution of the following:
sin θ = `-1/sqrt(2)`
Solve the following equations:
cos 2θ = `(sqrt(5) + 1)/4`
Choose the correct alternative:
If f(θ) = |sin θ| + |cos θ| , θ ∈ R, then f(θ) is in the interval
If a cosθ + b sinθ = m and a sinθ - b cosθ = n, then show that a2 + b2 = m2 + n2
