Could a trig expert please help me with this math problem?

Question

A plane flies in a straight line at 500 mph for 1 hour and 28 minutes. It makes a 12° turn and flies at 450 mph for 2 hours and 23 minutes. How far is it from its starting point (in miles)?

Answer 1

the trajectory forms an obtuse triangle with the following distances as arms:

notice that distance = rate x time

A = 500 x [1+ (28/60)] = 733.3mi
B = 450 x [2+ (23/60)] = 1072.5 mi
c = 12
C = unknown

using the law of cosines, solve for C

C = sqrt [A^2 + B^2 + 2AB cos(c)]
C = sqrt [733.3^2 + 1072.5^2 + 2(733.3)(1072.5)cos(12)]
C = 1796.3 mi

Answer 2

Plane starts from A

After 1 hour and 28 minutes the plane is 500*(1+28/60)= 733 1/3 miles from A --- Call this point B

From point B:
After a further 2 hours and 23 minutes, the plane is 450*(2+23/60) = 1072.5 miles from point B

Distance in original direction
= 733 1/3 + 1072.5cos12 = u

Distance perpendicular to this direction
= 1072.5sin12 = v

Total distance
= sqrt(u^2+v^2)
= sqrt[733.33^2 + 2*733.33*1072.5cos12 + (1072.5^2)cos^2(12) + (1072.5^2)sin^2 (2)]
= sqrt(733.33^2 + 1072.5^2 + 2*733.33*1072.5cos12)
= 1796.3 miles (approx)

Answer 3

This is equivalent to finding the third side of a triangle given two sides and the included angle.

The include angle is 180 - 12 = 168°.

a = 500(88/60) = 2200/3
b = 450(143/60) = 2145/2

Now use the Law of Cosines.

c² = a² + b² - 2ab cosC
c² = (2200/3)² + (2145/2)² - 2*(2200/3)*(2145/2) cos168°
c ≈ 1796.2907 miles

Answer 4

via periodic nature of the sine and cosine purposes. x = ok*? you may make the expression on the left equivalent to 0. whilst is sinx = 0 ? Sin(ok*?) ok = 0 Sin0 = 0 ok = a million sin? = 0 ok=2 sin2? = 0 etc.... As for cosx cos(ok*?) whilst it fairly is comparable to a million the expression turns into 0 via (a million-cosx) So ok=0 cos(0) = a million => a million-a million = 0 ok=a million cos? = -a million => a million+a million = 2 (does not artwork, yet is roofed via the sin expression) ok=a million cos2? = a million etc... The action repeats it fairly is self each 360 stages or 2? rads -2sin(ok*?)*( a million-cos(ok*?) ) = 0, for ok ? integers.