Question

the relationship between the cones vertex angle, θ, 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 = 3/2, determine the angle θ. sin(θ/2) = 1/m θ = □° (round to the nearest degree as needed.)

Explanation:

Step1: Substitute the value of \(m\) into the formula

Given \(m = \frac{3}{2}\), substitute it into \(\sin\frac{\theta}{2}=\frac{1}{m}\). So we have \(\sin\frac{\theta}{2}=\frac{1}{\frac{3}{2}}\).
Using the rule of dividing by a fraction (\(a\div\frac{b}{c}=a\times\frac{c}{b}\)), \(\frac{1}{\frac{3}{2}} = 1\times\frac{2}{3}=\frac{2}{3}\). So \(\sin\frac{\theta}{2}=\frac{2}{3}\).

Step2: Solve for \(\frac{\theta}{2}\)

We know that if \(\sin x = y\), then \(x=\sin^{- 1}(y)\). Let \(x = \frac{\theta}{2}\) and \(y=\frac{2}{3}\). So \(\frac{\theta}{2}=\sin^{-1}(\frac{2}{3})\).
Using a calculator, \(\sin^{-1}(\frac{2}{3})\approx41.81^{\circ}\).

Step3: Solve for \(\theta\)

Since \(\frac{\theta}{2}\approx41.81^{\circ}\), then \(\theta = 2\times41.81^{\circ}\).
\(\theta\approx83.62^{\circ}\). Rounding to the nearest degree, \(\theta\approx84^{\circ}\).

Answer:

\(84\)