The relationship between the cone's vertex angle, $\theta$, and the Mach number, $m$, of an aircraft that is flying faster than the speed of sound is given by the formula below. If $m=\frac{3}{2}$, determine the angle $\theta$.
\[
\sin \frac{\theta}{2}=\frac{1}{m}
\]
\[
\theta=
\]
(Round to the nearest degree as needed.)
Answer
Final Answer: The angle \(\theta\) is \(\boxed{84}\) degrees.
Step 1 ：Given the Mach number, m = 1.5, we can substitute this into the given formula: \(\sin \frac{\theta}{2}=\frac{1}{m}\).
Step 2 ：Taking the inverse sine of both sides of the equation to isolate \(\theta\), we get: \(\theta = 2 \sin^{-1}(\frac{1}{m})\).
Step 3 ：Substituting the given value of m into the equation, we get: \(\theta = 2 \sin^{-1}(\frac{1}{1.5})\).
Step 4 ：Calculating the inverse sine and multiplying by 2, we get: \(\theta = 84\) degrees.
Step 5 ：Rounding to the nearest degree, we get: \(\theta = 84\) degrees.
Step 6 ：Final Answer: The angle \(\theta\) is \(\boxed{84}\) degrees.
