Advertisements
Advertisements
प्रश्न
Solve the equation for x: `sin^(-1) 5/x + sin^(-1) 12/x = π/2, x ≠ 0`
Solve the equation x: `sin^(-1) (5/x) + sin^(-1) (12/x) = π/2 (x ≠ 0)`
Advertisements
उत्तर १
`sin^(-1) (5/x) + sin^(-1) (12/x) = π/2`
`sin^(-1) + cos^(-1) sqrt(1 - 144/x^2) = π/2`
Let `sin^(-1) 12/x = β`
`12/x = sin β = "OPP"/"HYP"`
`sqrt(x^2 - 144)/x = cos β = "Adj"/"HYP"`
`β = cos^(-1) (sqrt(x^2 - 144)/x^2)`
∴ `5/x = sqrt(1 - 144/x^2)`
`25/x^2 = 1 - 144/x^2`
`169/(x^2) = 1`
x2 = 169
x = 13
उत्तर २
`sin^(-1) (5/x) + sin^(-1) (12/x) = π/2`
⇒ `sin^-1 (5/x) = π/2 - sin^-1 (12/x)`
⇒ `sin^-1 (5/x) = cos^-1 (12/x)` ...`[∵ sin^-1x + cos^-1x = π/2]`
⇒ `sin^-1 (5/x) = cos^-1 (12/x) = θ`
⇒ `sin θ = 5/x, cos θ = 12/x`
Since, sin2θ + cos2θ = 1
⇒ `(5/x)^2 + (12/x)^2 = 1`
⇒ `25/(x^2) + 144/(x^2) = 1`
⇒ `169/(x^2) = 1`
⇒ x = ±13
Since, x = –13 does not satisfy the given equation.
So, x = 13.
Notes
Students should refer to the answer according to their questions.
संबंधित प्रश्न
Verify that y = 4 sin 3x is a solution of the differential equation \[\frac{d^2 y}{d x^2} + 9y = 0\]
Show that y = ex (A cos x + B sin x) is the solution of the differential equation \[\frac{d^2 y}{d x^2} - 2\frac{dy}{dx} + 2y = 0\]
Differential equation \[\frac{dy}{dx} = y, y\left( 0 \right) = 1\]
Function y = ex
Differential equation \[\frac{d^2 y}{d x^2} + y = 0, y \left( 0 \right) = 0, y' \left( 0 \right) = 1\] Function y = sin x
(sin x + cos x) dy + (cos x − sin x) dx = 0
(y2 + 1) dx − (x2 + 1) dy = 0
Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y + 2 (x + 1) = 2e2x.
The tangent at any point (x, y) of a curve makes an angle tan−1(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).
The integrating factor of the differential equation (x log x)
\[\frac{dy}{dx} + y = 2 \log x\], is given by
The solution of the differential equation \[\frac{dy}{dx} = \frac{ax + g}{by + f}\] represents a circle when
Which of the following transformations reduce the differential equation \[\frac{dz}{dx} + \frac{z}{x}\log z = \frac{z}{x^2} \left( \log z \right)^2\] into the form \[\frac{du}{dx} + P\left( x \right) u = Q\left( x \right)\]
The integrating factor of the differential equation \[x\frac{dy}{dx} - y = 2 x^2\]
Solve the following differential equation.
`dy/dx = x^2 y + y`
For each of the following differential equations find the particular solution.
`y (1 + logx)dx/dy - x log x = 0`,
when x=e, y = e2.
Solve the following differential equation.
x2y dx − (x3 + y3) dy = 0
Solve the following differential equation.
`x^2 dy/dx = x^2 +xy - y^2`
Solve:
(x + y) dy = a2 dx
Solve the differential equation xdx + 2ydy = 0
Solve: ydx – xdy = x2ydx.
The differential equation (1 + y2)x dx – (1 + x2)y dy = 0 represents a family of:
Solve the differential equation `dy/dx + xy = xy^2` and find the particular solution when y = 4, x = 1.
