In aviation there is a thumb rule called the 1 in 60 rule Re

In aviation, there is a thumb rule called the 1 in 60 rule. Recall from MA140G that a thumb rule is a linearization that allows effective estimation in a dynamic, multi-tasked environment. The 1 in 60 rule states that for every 60 miles traveled, every 1 mile deviation off course is about 1 degree in heading error. a. Show that this rule is approximately true by doing a simple trig calculation. b. If a pilot has flown about 180 miles and then determines she is 9 miles off course by pilotage (looking out the window at a feature), use the 1 in 60 rule to estimate how many degrees she should correct to align parallel to her intended course. c. From part b, calculate the exact number of degrees she should correct. How many miles would she need to be off-track before the 1 in 60 rule would be more than 2 degrees in error? d. This thumb rule uses a linearization of trig functions called a small sine approximation. This approximates trig functions for small angles (typically only a few degrees). Considering a small central angle in the unit circle, what do the cosine and sine functions approximately equal, respectively? Explain how this approximation is being used in this rule.

Solution

a) As per the diagram given apply trig, formula

tan(theta) = 1/60

theta = tan^-1(1/60) = 0.9548 = 1 degree approx.

b) As per 1 in 60 rule:

For 180 mile travelled , she should correct 3 degees to align parallel to intended course

c) by trig function : tan(theta) = 9/180

theta = 2.86 deg is the exact number of degrees she should correct.

Lets assume she is x miles off track , then : tan^-1(x/180) > 2deg

x/180 > 0.0349

x > 6.28 miles off track

d) for example for very small theta = sintheta = costheta

So, tan(theta) = theta = 1/60 = 0.0167 rad ;

in degrees = 180*0.0167/pi = 0.954 = 1 degree