જો બે વર્તુળોમાં સમાન લંબાઈનાં ચાપ કેન્દ્ર આગળ $60^{\circ}$ અને $75^{\circ}$ ના ખૂણા આંતરે, તો તેમની ત્રિજ્યાઓનો ગુણોત્તર શોધો.
Let the radii of the two circles be $r_{1}$ and $r_{2} .$ Let an arc of length $l$ subtend an angle of $60^{\circ}$ at the centre of the circle of radius $r_{1},$ while let an arc of length/subtend an angle of $75^{\circ}$ at the centre of the circle of radius $r_{2}$
Now, $60^{\circ}=\frac{\pi}{3}$ radian and $75^{\circ}=\frac{5 \pi}{12}$ radian
We know that in a circle of radius $r$ unit, if an arc of length $l$ unit subtends an angle $\theta$ radian at the centre then
$\theta=\frac{l}{r}$ or $l=r \theta$
$\therefore l=\frac{r_{1} \pi}{3}$ and $l=\frac{r_{2} 5 \pi}{12}$
$\Rightarrow \frac{r_{1} \pi}{3}=\frac{r_{2} 5 \pi}{12}$
$\Rightarrow r_{1}=\frac{r_{2} 5}{4}$
$\Rightarrow \frac{r_{1}}{r_{2}}=\frac{5}{4}$
Thus, the ratio of the radii is $5: 4 $
$\sin \frac{x}{2}, \cos \frac{x}{2}$ અને $\tan \frac{x}{2}$ ની કિંમતો શોધો.: $\cos x=-\frac{1}{3}, x$ એ બીજા ચરણમાં છે.
જો ${\tan ^2}\alpha {\tan ^2}\beta + {\tan ^2}\beta {\tan ^2}\gamma + {\tan ^2}\gamma {\tan ^2}\alpha $ $ + 2{\tan ^2}\alpha {\tan ^2}\beta {\tan ^2}\gamma = 1,$ તો ${\sin ^2}\alpha + {\sin ^2}\beta + {\sin ^2}\gamma =.........$
$\sin \left( {\frac{\pi }{{10}}} \right)\sin \left( {\frac{{3\pi }}{{10}}} \right) = $
$(m + 2)\sin \theta + (2m - 1)\cos \theta = 2m + 1,$ જો . . .
જો $\sin \theta + \cos \theta = 1$, તો $\sin \theta \cos \theta = $