If sin y = x sin (a + y), prove that dy dx = ^ 2 (a + y) . - Vidyadip

Question

If sin y = x sin (a + y), prove that $\frac{\text{dy}}{\text{dx}}=\frac{\sin^{2}\text{(a + y)}}{\sin\text{a}}.$

✓

Answer

$\sin\text{y}=\text{x}\sin\text{ (a + y)}\Rightarrow\cos\text{y }\frac{\text{dy}}{\text{dx}}=\sin\text{(a + y)}+\text{x}\cos\text{(a + y)}\frac{\text{dy}}{\text{dx}}........\text{(i)}$

$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{sin (a + y)}}{\cos\text{ y - x }\cos\text{(a + y)}}$

From (i), x = $\frac{\sin\text{y}}{\sin\text{(a + y)}}\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{sin (a + y)}}{\cos\text{y}-\frac{\sin\text{y}}{\sin\text{(a + y)}}\cdot\cos\text{(a + y)}}$

$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\sin^{2}\text{(a + y)}}{\sin(\text{a + y - y)}}=\frac{\sin^{2}\text{(a + y)}}{\sin\text{a}}.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,
f(x) = (x - 1)(x - 2)²

Find the equatoion of the passing through the points (1, -1, 2) and (2, -2, 2) and which is perpendicular to the plane 6x - 2y + 2z = 9.

Solve the following linear programming problem by graphical method. Under the following constraints :
$
\begin{aligned}
x+2 y & \geq 10 \\
x+y & \geq 6 \\
3 x+y & \geq 8 \\
x, y & \geq 0
\end{aligned}
$
$\operatorname{minimise} Z=3 x+5 y$.

A fair die is tossed. Let X denote 1 or 3 according as an odd or an even number appears. Find the probability distribution, mean and variance of X.

Evaluate the following:
$\begin{vmatrix}\text{x}&1&1\\1&\text{x}&1\\1&1&\text{x}\end{vmatrix}$

Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-1}{3}=\frac{\text{y}-1}{-1}=\frac{\text{z}+1}{0}$ and $\frac{\text{x}-4}{2}=\frac{\text{y}-0}{0}=\frac{\text{z}+1}{3}$

Integrate the following integrals:
$\int\sin2\text{x}\sin4\text{x}\sin6\text{x dx}$

Using integeration, find the area of the region bounded by the y - 1 = x, the x-axis and the ordinates x = -2 and x = 3.

The random variable X can take only the values 0, 1, 2, 3. Given that P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) such that $\sum{\text{p}}_{\text{i}}x^{2}_{\text{i}} = 2\sum\text{p}_{\text{i}}x_{\text{i}},$find the value of p.

Compute the elements a₄₃ and a₂₂ of the matrix:
$\text{A}=\begin{bmatrix}0&1&0\\2&0&2\\0&3&2\\4&0&4\end{bmatrix}\begin{bmatrix}2&-1\\-3&2\\4&3\end{bmatrix}\begin{bmatrix}0&1&-1&2&-2\\3&-3&4&-4&0\end{bmatrix}$