If critical frequency is 20 MHz for ionosphere, calculate the angle of transmission if 30 MHz frequency is transmitted.

Options

A. `cos ^(-1)((2)/(3))`
B. `sin ^(-1)((2)/(3))`
C. `cos ^(-1)((1)/(3))`
D. `sin ^(-1)((1)/(3))`

Answer

To solve the problem, we need to find the angle of transmission when a frequency of 30 MHz is transmitted, given that the critical frequency for the ionosphere is 20 MHz. We can use the relationship between the maximum frequency transmitted, the critical frequency, and the angle of transmission.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Critical frequency, \( f_c = 20 \, \text{MHz} \)
- Maximum transmitted frequency, \( f_{max} = 30 \, \text{MHz} \)

2. **Use the Relationship Between Frequencies and Angle:**
The relationship between the maximum frequency transmitted, the critical frequency, and the angle of transmission \( \theta \) is given by the formula:
\[ f_{max} = \frac{f_c}{\cos \theta} \]

3. **Rearrange the Formula to Solve for \( \cos \theta \):**
To find \( \cos \theta \), we can rearrange the formula:
\[ \cos \theta = \frac{f_c}{f_{max}} \]

4. **Substitute the Given Values:**
Now, substitute the values of \( f_c \) and \( f_{max} \):
\[ \cos \theta = \frac{20 \, \text{MHz}}{30 \, \text{MHz}} = \frac{20}{30} = \frac{2}{3} \]

5. **Calculate the Angle \( \theta \):**
To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{2}{3}\right) \]

6. **Final Answer:**
The angle of transmission \( \theta \) is:
\[ \theta = \cos^{-1}\left(\frac{2}{3}\right) \]