Question

Solving a realistic word problem using trig identities? See details.

The speed of a supersonic aircraft is usually represented by a Mach number, named after Austrian physicist Ernst Mach (1838–1916). A Mach number is the speed of the aircraft, in miles per hour, divided by the speed of sound, approximately 740 miles per hour. Thus, a plane flying at twice the speed of sound has a speed, M, of Mach 2. If an aircraft has a speed greater than Mach 1, a sonic boom is heard, created by sound waves that form a cone with a vertex angle u, shown in the figure. The relationship between the cone’s vertex angle, u, and the Mach speed, M, of an aircraft that is flying faster than the speed of sound is given by sin(theta/2) = 1/M

If theta = pi/6 , determine the Mach speed, M, of the aircraft.
Express the speed as an exact value and as a decimal to the nearest tenth.

Answer 1

Something that might help

Explanation:

`sin(theta/2) = 1/M`

`sin(pi/12) = sin((pi/6)/2) = 1/M`

Using the double angles theorem,

`sqrt((1-cos(A))/2)=sin(A/2)`

Now plugging in,

`sqrt((1-cos(pi/6))/2)=sin((pi/6)/2)`

`sqrt((1-sqrt3/2)/2)=1/M`

`sqrt((1/2-sqrt3/4))=1/M`

`sqrt((2/4-sqrt3/4))=1/M`

`sqrt(((2-sqrt3)/4))=1/M`

`sqrt(2-sqrt3)/2=1/M` or `(2-sqrt3)/4=1/M^2`

`2/(2-sqrt3)^(1/2)=M`or or `4/(2-sqrt3)=M^2`

`2-4/sqrt3 = M^2`

`2-(4sqrt3)/3=M^2`

`6/3-(4sqrt3)/3=M^2`

`(6-4sqrt3)/3=M^2`

Depending on the wording of the problem, I think you have to divide or multiply by 740 after somehow eliminating the square root.

And it will (might) require a calculator
(Or you could have just skipped a lot of the first steps and put in the original equation into a calculator...?)

I don't actually have an answer