If the arcs of the same lengths in two circles subtend angles $65^{\circ}$ and $110^{\circ}$ at the centre, find the ratio of their radii.
Let $r_{1}$ and $r_{2}$ be the radii of the two circles. Given that
${{\theta _1} = {{65}^\circ } = \frac{\pi }{{180}} \times 65 = \frac{{13\pi }}{{36}}\,{\text{ radian }}}$
and ${{\theta _2} = {{110}^\circ } = \frac{\pi }{{180}} \times 110 = \frac{{22\pi }}{{36}}{\text{ }}\,{\text{radian }}}$
Let $l$ be the length of each of the arc. Then $l=r_{1} \theta_{1}=r_{2} \theta_{2},$ which gives
$\frac{13 \pi}{36} \times r_{1}=\frac{22 \pi}{36} \times r_{2}, \text { i.e., } \frac{r_{1}}{r_{2}}=\frac{22}{13}$
Hence $r_{1}: r_{2}=22: 13$
If $\tan \theta = - \frac{1}{{\sqrt {10} }}$ and $\theta $ lies in the fourth quadrant, then $\cos \theta = $
Find the value of the trigonometric function $\sin 765^{\circ}$
If $x = \sec \theta + \tan \theta ,$ then $x + \frac{1}{x} = $
If $x = \sec \,\phi - \tan \phi ,y = {\rm{cosec}}\phi + \cot \phi ,$ then
Let the function $:(0, \pi) \rightarrow R$ be defined by
$f (\theta)=(\sin \theta+\cos \theta)^2+(\sin \theta-\cos \theta)^4$
Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in\left\{\lambda_1 \pi, \ldots, \lambda_{ T } \pi\right\}$, where $0<\lambda_1<\cdots<\lambda_r<1$. Then the value of $\lambda_1+\cdots+\lambda_r$ is. . . . .