4. What is a reasonabie approximation for anele fif the ratio of the affacerit leg Chitect ty the hypotentse is \( 0 . \mathrm{a}^{9} \) ? A. 27 aegrees A. 30 degrees C. 60 degrees D. 63 degrees

To solve the problem, we need to find the angle \( \theta \) in a right triangle where the ratio of the opposite leg to the hypotenuse is given as \( 0.a^9 \). This ratio corresponds to the sine of the angle \( \theta \): \[ \sin(\theta) = 0.a^9 \] Since \( 0.a^9 \) is a very small number, we can approximate the angle using the sine function. 1. **Understanding the sine function**: The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. For small angles, the sine of the angle is approximately equal to the angle itself when measured in radians. 2. **Finding the angle**: We can use the inverse sine function to find the angle \( \theta \): \[ \theta = \arcsin(0.a^9) \] 3. **Estimating the value**: Since \( 0.a^9 \) is a small value, we can use the fact that for small angles (in radians), \( \sin(\theta) \approx \theta \). Now, let's convert the approximate sine value to degrees. Given the options: - A. 27 degrees - B. 30 degrees - C. 60 degrees - D. 63 degrees Since \( 0.a^9 \) is a small value, we can estimate that the angle will be less than 30 degrees. Thus, the most reasonable approximation for the angle \( \theta \) is: **Answer: A. 27 degrees**.