3.Trigonometrical Ratios, Functions and Identities
normal

If the arcs of the same length in two circles $S_1$ and $S_2$ subtend angles $75^o $ and $120^o $ respectively at the centre. The ratio $\frac{{{S_1}}}{{{S_2}}}$ is equal to

A

$\frac{1}{5}$

B

$\frac{{81}}{{16}}$

C

$\frac{{64}}{{25}}$

D

$\frac{{25}}{{64}}$

Solution

$\theta=\frac{l}{r}$

$\theta \rightarrow$ angle (in radians)

$l \rightarrow$ are length

$r \rightarrow$ radius

$l_{1}=l_{2}=l(\operatorname{say})$

$\theta_{1}=75^{\circ} \quad ; \quad \theta_{2}=120^{\circ}$

$1^{0}=\left(\frac{\pi}{180}\right)^{c}$

$\theta_{1}=75^{\circ}=\left(\frac{5 \pi}{12}\right)^{\text {radians }}$

$\theta_{1}=120^{\circ}=\left(\frac{2 \pi}{3}\right)^{\text {radians }}$

$\frac{r_{1}}{r_{2}}=\frac{\frac{l_{1}}{\theta_{1}}}{\frac{\lambda_{2}}{\theta_{2}}}$

$=\frac{l_{1}}{\theta_{1}} \times \frac{\theta_{2}}{l_{2}}$

$r_{1}: r_{2}=8: 5$

Std 11
Mathematics

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