Answer:
a) θ = π
b) θ = 24°
c) 760 miles /hour , 3800 miles/hour
d) sin α / 2√ (cos α + 1 )/2*
Step-by-step explanation:
a) If 1 is a mach number then 1/M = 1/1 = 1
sin (θ/2) = 1/M ⇒ sin (θ/2) = 1 ⇒ arcsin (1) = π/2
Then (θ/2) = π/2 ⇒ θ = π
b) If 5 is a mach number then 1/M = 1/5 = 0.2
sin (θ/2) = 1/5 ⇒ arcsin (0.2)
θ/2 = 12° ⇒ θ = 24°
c) speed of sound is 760 miles/hour
speed of an object with mach number 1 is 760 miles /hour
speed of an object with mach number 5 is 3800 miles/hour
d) We know:
sin2α = 2 sinα cosα or sin α = 2 sinα/2 cosα/2
sinα/2 = sin α / 2*cos α/2 (1)
And cos 2α = 2cos² α/2 - 1 then cosα = 2* cos ²α/2 - 1
cos ²α/2 = ( cosα + 1 )/2 ⇒ cos α/2 =√ (cos α + 1 )/2
Plugging that value in equation 1 we get
sinα/2 = sin α / 2√ (cos α + 1 )/2*