The mach number M of an airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane (see figure). The mach number is related to the apex angle θ of the cone by the equation below.
Sin(θ/2)=1/M

(a) Find the angle θ that corresponds to a mach number of 1.
θ= degrees
(b) Find the angle θ that corresponds to a mach number of 5.
θ= degrees
(c) The speed of sound is about 760 miles per hour. Determine the speed of an object having the mach numbers from parts (a) and (b).
Mach Number of 1 mph
Mach Number of 5 mph
(d) Rewrite the equation in terms of θ

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*

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step-by-step explanation: