Advertisements
Advertisements
Question
Solve the following equation:
\[5 \cos^2 x + 7 \sin^2 x - 6 = 0\]
Advertisements
Solution
\[5 \cos^2 x + 7 \sin^2 x - 6 = 0\]
\[ \Rightarrow 5 \cos^2 x + 7\left( 1 - \cos^2 x \right) - 6 = 0\]
\[ \Rightarrow - 2 \cos^2 x + 1 = 0\]
\[ \Rightarrow \cos^2 x = \frac{1}{2} = \cos^2 \frac{\pi}{4}\]
\[ \Rightarrow x = n\pi \pm \frac{\pi}{4}, n \in Z \left( \cos^2 x = \cos^2 \alpha \Rightarrow x = n\pi \pm \alpha, n \in Z \right)\]
APPEARS IN
RELATED QUESTIONS
Find the principal and general solutions of the equation `cot x = -sqrt3`
Find the general solution of the equation cos 3x + cos x – cos 2x = 0
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 \[a = \sec x - \tan x \text{ and }b = cosec x + \cot x\], then shown that \[ab + a - b + 1 = 0\]
Prove that:
Prove that:
\[\frac{\cos (2\pi + x) cosec (2\pi + x) \tan (\pi/2 + x)}{\sec(\pi/2 + x)\cos x \cot(\pi + x)} = 1\]
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:
\[\tan 4\pi - \cos\frac{3\pi}{2} - \sin\frac{5\pi}{6}\cos\frac{2\pi}{3} = \frac{1}{4}\]
If tan x = \[x - \frac{1}{4x}\], then sec x − tan x is equal to
If \[\frac{\pi}{2} < x < \frac{3\pi}{2},\text{ then }\sqrt{\frac{1 - \sin x}{1 + \sin x}}\] is equal to
If \[cosec x - \cot x = \frac{1}{2}, 0 < x < \frac{\pi}{2},\]
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:
\[\sqrt{3} \cos x + \sin x = 1\]
Solve the following equation:
\[2^{\sin^2 x} + 2^{\cos^2 x} = 2\sqrt{2}\]
If cos x = k has exactly one solution in [0, 2π], then write the values(s) of k.
Write the number of points of intersection of the curves
Write the values of x in [0, π] for which \[\sin 2x, \frac{1}{2}\]
and cos 2x are in A.P.
If \[\cos x + \sqrt{3} \sin x = 2,\text{ then }x =\]
In (0, π), the number of solutions of the equation \[\tan x + \tan 2x + \tan 3x = \tan x \tan 2x \tan 3x\] is
General solution of \[\tan 5 x = \cot 2 x\] is
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
sin4x = sin2x
Solve the following equations for which solution lies in the interval 0° ≤ θ < 360°
2 cos2x + 1 = – 3 cos x
Solve the following equations:
sin 5x − sin x = cos 3
Solve the following equations:
`sin theta + sqrt(3) cos theta` = 1
Choose the correct alternative:
If cos pθ + cos qθ = 0 and if p ≠ q, then θ is equal to (n is any integer)
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
Find the general solution of the equation sinx – 3sin2x + sin3x = cosx – 3cos2x + cos3x
The minimum value of 3cosx + 4sinx + 8 is ______.
