The graph between sine of angle of refraction $(\sin r)$ in medium 2 and sine of angle of incidence $(\sin i)$ in medium 1 indicates that $\left(\tan 36^{\circ}=\frac{3}{4}\right)$.

A. Total internal reflection can take place
B. Total internal reflection cannot take place
C. Any of a and b
D. Data is incomplete

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Given the problem, we are provided with a graph depicting the relationship between $\sin r$ (sine of the angle of refraction in medium 2) and $\sin i$ (sine of the angle of incidence in medium 1). The problem also specifies that $\tan 36^{\circ} = \frac{3}{4}$.

We need to determine if total internal reflection can take place by analyzing the provided graph.


  1. Graph Analysis:
    The graph shows the relationship between $\sin r$ and $\sin i$. From the graph, we can deduce the slope which is given by the angle $\frac{2\pi}{10} = 36^{\circ}$.

  2. Calculating the Refractive Index:
    Using the slope of the line: $$ \tan 36^{\circ} = \frac{\sin i}{\sin r} $$ Given $\tan 36^{\circ} = \frac{3}{4}$, we find: $$ \frac{\sin i}{\sin r} = \frac{3}{4} $$

  3. Applying Snell's Law:
    According to Snell's Law: $$ \frac{\sin i}{\sin r} = \frac{n_2}{n_1} $$ where $n_2$ and $n_1$ are the refractive indices of medium 2 and medium 1, respectively.

    From the given ratio, we equate: $$ \frac{n_2}{n_1} = \frac{3}{4} $$

    This means: $$ n_2 = \frac{3}{4} n_1 $$

  4. Relative Refractive Indices:
    Given that $n_2 < n_1$, medium 2 is less optically dense than medium 1.

  5. Understanding Total Internal Reflection (TIR):
    Total internal reflection occurs only when light travels from a denser to a rarer medium. Here, since medium 2 is less dense than medium 1, total internal reflection is not possible.


Total internal reflection cannot take place. Therefore, the correct answer is:

B. Total internal reflection cannot take place.

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