Advertisements
Advertisements
Question
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.
Advertisements
Solution
2sin2θ = 3cosθ
We know that,
sin2θ = 1 – cos2θ
Given that,
2sin2θ = 3 cosθ
2 – 2cos2θ = 3cosθ
2cos2θ + 3cosθ – 2 = 0
(cosθ + 2)(2cosθ – 1) = 0
Therefore,
cosθ = `1/2 = cos pi/3`
θ = `pi/3` or `2π – pi/3`
θ = `pi/3, (5pi)/3`
Therefore, 2(1 – cos2θ) = 3cosθ
⇒ 2 – 2cos2θ = 3cosθ
⇒ 2cos2θ + 3cosθ – 2 = 0
⇒ 2cos2θ + 4cosθ – cosθ – 2 = 0
⇒ 2cosθ(cosθ + 2) + 1(cosθ + 2) = 0
⇒ (2cosθ + 1)(cosθ + 2) = 0
Since, cosθ ∈ [–1, 1], for any value θ.
So, cosθ ≠ –2
Therefore,
2cosθ – 1 = 0
⇒ cosθ = `1/2`
= `pi/3` or `2π – pi/3`
θ = `π/3, (5pi)/3`
APPEARS IN
RELATED QUESTIONS
Find the principal and general solutions of the equation `cot x = -sqrt3`
Find the general solution for each of the following equations sec2 2x = 1– tan 2x
If \[\sin x = \frac{a^2 - b^2}{a^2 + b^2}\], then the values of tan x, sec x and cosec x
If \[\tan x = \frac{b}{a}\] , then find the values of \[\sqrt{\frac{a + b}{a - b}} + \sqrt{\frac{a - b}{a + b}}\].
Prove that:
Prove that
Prove that:
\[\sin^2 \frac{\pi}{18} + \sin^2 \frac{\pi}{9} + \sin^2 \frac{7\pi}{18} + \sin^2 \frac{4\pi}{9} = 2\]
Prove that:
\[\sec\left( \frac{3\pi}{2} - x \right)\sec\left( x - \frac{5\pi}{2} \right) + \tan\left( \frac{5\pi}{2} + x \right)\tan\left( x - \frac{3\pi}{2} \right) = - 1 .\]
In a ∆ABC, prove that:
Prove that:
Prove that:
If \[\frac{\pi}{2} < x < \frac{3\pi}{2},\text{ then }\sqrt{\frac{1 - \sin x}{1 + \sin x}}\] is equal to
If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then x2 + y2 + z2 is independent of
If tan A + cot A = 4, then tan4 A + cot4 A is equal to
If \[f\left( x \right) = \cos^2 x + \sec^2 x\], then
Find the general solution of the following equation:
Find the general solution of the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
Solve the following equation:
\[5 \cos^2 x + 7 \sin^2 x - 6 = 0\]
If secx cos5x + 1 = 0, where \[0 < x \leq \frac{\pi}{2}\], find the value of x.
Write the number of solutions of the equation
\[4 \sin x - 3 \cos x = 7\]
Write the number of points of intersection of the curves
If \[3\tan\left( x - 15^\circ \right) = \tan\left( x + 15^\circ \right)\] \[0 < x < 90^\circ\], find θ.
If \[\tan px - \tan qx = 0\], then the values of θ form a series in
If \[\cot x - \tan x = \sec x\], then, x is equal to
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
The equation \[3 \cos x + 4 \sin x = 6\] has .... solution.
General solution of \[\tan 5 x = \cot 2 x\] is
If sin θ and cos θ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation ______.
Find the general solution of the equation sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x
