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 $\sin {\theta _1} + \sin {\theta _2} + \sin {\theta _3} = 3,$ then $\cos {\theta _1} + \cos {\theta _2} + \cos {\theta _3} = $
If $\theta $ lies in the second quadrant, then the value of $\sqrt {\left( {\frac{{1 - \sin \theta }}{{1 + \sin \theta }}} \right)} + \sqrt {\left( {\frac{{1 + \sin \theta }}{{1 - \sin \theta }}} \right)} $
$\frac{{\sin \theta }}{{1 - \cot \theta }} + \frac{{\cos \theta }}{{1 - \tan \theta }} = $
Find the values of other five trigonometric functions if $\sec x=\frac{13}{5}, x$ lies in fourth quadrant.
$\cos 1^\circ + \cos 2^\circ + \cos 3^\circ + ..... + \cos 180^\circ = $