Advertisements
Advertisements
प्रश्न
Solve the following equation:
`cosec x = 1 + cot x`
Advertisements
उत्तर
Given,
`cosec x = 1 + cot x`
⇒ `1/sin x = 1 + cos x/sin x`
⇒ sin x + cos x = 1
In all such problems we try to reduce the equation in an equation involving single trigonometric expression.
∴ `s 1/sqrt2 sin x + 1/sqrt2 cos x = 1/sqrt2` {dividing by √2 both sides}
⇒ `sin x sin pi/4 + cos pi/4 cos x = cos pi/4.` {cos A cos B + sin A sin B = cos(A − B)}
NOTE: The ratio of sin can also be used in place of cos; the answer stays the same, but the form may change. If you enter numbers for n, you will receive the same values in both forms.
If cos x = cos y, impls x = 2nπ ± y, where n ∈ Z
∴ `x - pi/4 = (2npi ± pi/4).`
∴ `x = (2npi ± pi/4) + pi/4` where n n ∈ Z
`x = 2npi or x = 2npi + pi/2` where n n ∈ Z
APPEARS IN
संबंधित प्रश्न
If \[cosec x - \sin x = a^3 , \sec x - \cos x = b^3\], then prove that \[a^2 b^2 \left( a^2 + b^2 \right) = 1\]
If \[\sin x + \cos x = m\], then prove that \[\sin^6 x + \cos^6 x = \frac{4 - 3 \left( m^2 - 1 \right)^2}{4}\], where \[m^2 \leq 2\]
If \[a = \sec x - \tan x \text{ and }b = cosec x + \cot x\], then shown that \[ab + a - b + 1 = 0\]
If \[T_n = \sin^n x + \cos^n x\], prove that \[6 T_{10} - 15 T_8 + 10 T_6 - 1 = 0\]
Prove that:
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:
\[\sin^2 \frac{\pi}{18} + \sin^2 \frac{\pi}{9} + \sin^2 \frac{7\pi}{18} + \sin^2 \frac{4\pi}{9} = 2\]
In a ∆ABC, prove that:
Find x from the following equations:
\[x \cot\left( \frac{\pi}{2} + \theta \right) + \tan\left( \frac{\pi}{2} + \theta \right)\sin \theta + cosec\left( \frac{\pi}{2} + \theta \right) = 0\]
If \[cosec x + \cot x = \frac{11}{2}\], then tan x =
Which of the following is incorrect?
Find the general solution of the following equation:
Find the general solution of the following equation:
Find the general solution of the following equation:
Solve the following equation:
\[\sin^2 x - \cos x = \frac{1}{4}\]
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:
Solve the following equation:
\[\sin x + \cos x = \sqrt{2}\]
Solve the following equation:
\[\sin x - 3\sin2x + \sin3x = \cos x - 3\cos2x + \cos3x\]
If cos x = k has exactly one solution in [0, 2π], then write the values(s) of k.
Write the values of x in [0, π] for which \[\sin 2x, \frac{1}{2}\]
and cos 2x are in A.P.
If \[\tan px - \tan qx = 0\], then the values of θ form a series in
The general solution of the equation \[7 \cos^2 x + 3 \sin^2 x = 4\] is
A value of x satisfying \[\cos x + \sqrt{3} \sin x = 2\] is
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
Solve the following equations:
`sin theta + sqrt(3) cos theta` = 1
Choose the correct alternative:
If f(θ) = |sin θ| + |cos θ| , θ ∈ R, then f(θ) is in the interval
Choose the correct alternative:
`(cos 6x + 6 cos 4x + 15cos x + 10)/(cos 5x + 5cs 3x + 10 cos x)` is equal to
Choose the correct alternative:
If sin α + cos α = b, then sin 2α is equal to
Solve `sqrt(3)` cos θ + sin θ = `sqrt(2)`
If 2sin2θ = 3cosθ, where 0 ≤ θ ≤ 2π, then find the value of θ.
Find the general solution of the equation 5cos2θ + 7sin2θ – 6 = 0
