Find an angle Which increases twice as fast as its cosine. - Vidyadip

Question

Find an angle $\theta$
Which increases twice as fast as its cosine.

✓

Answer

Let $\text{x}=\cos\theta$
Differentiating both sides with respect to t, we get
$\frac{\text{dx}}{\text{dt}}=\frac{\text{d}(\cos\theta)}{\text{dt}}$
$=-\sin\theta\frac{\text{d}\theta}{\text{dt}}$
But it is given that $\frac{\text{d}\theta}{\text{dt}}=2\frac{\text{dx}}{\text{dt}}$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=-\sin\theta\Big(2\frac{\text{dx}}{\text{dt}}\Big)$
$\Rightarrow\sin\theta=-\frac{1}{2}$
$\Rightarrow\theta=\pi+\frac{\pi}{6}=\frac{7\pi}{6}$
Hence, $\theta=\frac{7\pi}{6}.$

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 "3 Marks" from Derivative as a Rate Measurer→Derivative as a Rate Measurer — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

It the lines $\frac{\text{x}-1}{-3}=\frac{\text{y}-2}{2\lambda}=\frac{\text{z}-3}{2}$ and $\frac{\text{x}-1}{3\lambda}=\frac{\text{y}-2}{1}=\frac{\text{z}-6}{-5}$ are perpendicular, find the value of $\lambda.$

Find the volume of the parallelopiped whose coterminous edges are represented by the vectors:
$\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}},\vec{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}},\vec{\text{c}}=3\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}$

If $\theta$ is the angle between two unit vectors $\hat{a}$ and $\hat{b}$ then show that $\cos \frac{\theta}{2}=\frac{1}{2}|\hat{a}+\hat{b}|$.

Solve the following differential equations:

$\frac{\text{dy}}{\text{dx}}+\frac{\cos\text{x}\sin\text{y}}{\cos\text{y}}=0$

If y = 7x - x³ and x increases at the rate of 4 units per second, how fast is the slope of the curve changing when x = 2?

The radius of a circle is increasing at the rate of 0.7cm/ sec. What is the rate of increase of its circumfrence?

Find the particular solution of the differential equation $\frac { d y } { d x } - 3 y \cot x = \sin 2 x$ , given that y = 2 when $x = \frac { \pi } { 2 }$.

Three machines E₁, E₂, E₃ in a certain factory produce 50%, 25% and 25%, respectively, of the total daily output of electric bulbs. It is known that 4% of the tubes produced one each of the machines E₁and E₂ are defective, and that 5% of those produced on E₃ are defective. If one tube is picked up at random from a day's production, then calculate the probability that it is defective.

A piece of ice is in the form of a cube melts so that the percentage error in the edge of cube is a, then find the percentage error in its volume.

Evaluate the following integrals:

$\int\frac{1}{\text{x}\sqrt{4-9(\log\text{x})^2}}\text{ dx}$