Advertisements
Advertisements
Question
Solve `sqrt(3)` cos θ + sin θ = `sqrt(2)`
Advertisements
Solution
Divide the given equation by 2 to get
`sqrt(3)/2 cos theta + 1/2 sin theta = 1/sqrt(2)`
or `cos pi/6 cos theta + sin pi/6 sin theta = cos pi/4`
or `cos(pi/6 - theta) = cos pi/4` or `cos(theta - pi/6) = cos pi/4`
Thus, the solution is given by, i.e., θ = `2 m pi +- pi/4 + pi/6`
Hence, the solution is
θ = `2 m pi + pi/4 + pi/6` and `2 m pi - pi/4 + pi/6,` i.e., `theta = 2 m pi + (5pi)/12` and `theta = 2 m pi - pi / 12`
APPEARS IN
RELATED QUESTIONS
Find the principal and general solutions of the equation sec x = 2
If \[cosec x - \sin x = a^3 , \sec x - \cos x = b^3\], then prove that \[a^2 b^2 \left( a^2 + b^2 \right) = 1\]
If \[\cot x \left( 1 + \sin x \right) = 4 m \text{ and }\cot x \left( 1 - \sin x \right) = 4 n,\] \[\left( m^2 + n^2 \right)^2 = mn\]
Prove that: cos 24° + cos 55° + cos 125° + cos 204° + cos 300° = \[\frac{1}{2}\]
Prove that: tan (−225°) cot (−405°) −tan (−765°) cot (675°) = 0
Prove that
Prove that:
\[\sin^2 \frac{\pi}{18} + \sin^2 \frac{\pi}{9} + \sin^2 \frac{7\pi}{18} + \sin^2 \frac{4\pi}{9} = 2\]
In a ∆ABC, prove that:
In a ∆ABC, prove that:
Prove that:
\[\tan 4\pi - \cos\frac{3\pi}{2} - \sin\frac{5\pi}{6}\cos\frac{2\pi}{3} = \frac{1}{4}\]
Prove that:
\[\sin \frac{13\pi}{3}\sin\frac{2\pi}{3} + \cos\frac{4\pi}{3}\sin\frac{13\pi}{6} = \frac{1}{2}\]
If sec \[x = x + \frac{1}{4x}\], then sec x + tan x =
If \[0 < x < \frac{\pi}{2}\], and if \[\frac{y + 1}{1 - y} = \sqrt{\frac{1 + \sin x}{1 - \sin x}}\], then y is equal to
If \[\frac{3\pi}{4} < \alpha < \pi, \text{ then }\sqrt{2\cot \alpha + \frac{1}{\sin^2 \alpha}}\] is equal to
If tan A + cot A = 4, then tan4 A + cot4 A is equal to
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:
\[\sin x + \cos x = \sqrt{2}\]
Write the number of solutions of the equation tan x + sec x = 2 cos x in the interval [0, 2π].
Write the number of values of x in [0, 2π] that satisfy the equation \[\sin x - \cos x = \frac{1}{4}\].
If a is any real number, the number of roots of \[\cot x - \tan x = a\] in the first quadrant is (are).
The smallest positive angle which satisfies the equation
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
General solution of \[\tan 5 x = \cot 2 x\] is
Find the principal solution and general solution of the following:
tan θ = `- 1/sqrt(3)`
Solve the following equations:
2 cos2θ + 3 sin θ – 3 = θ
Solve the following equations:
sin θ + sin 3θ + sin 5θ = 0
If sin θ and cos θ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation ______.
If a cosθ + b sinθ = m and a sinθ - b cosθ = n, then show that a2 + b2 = m2 + n2
The minimum value of 3cosx + 4sinx + 8 is ______.
In a triangle ABC with ∠C = 90° the equation whose roots are tan A and tan B is ______.
