when is it beneficial to work with angles measured in radians and in degrees?

Answer 1

When dealing with strict two dimensional geometry... and never step outside of algebraic mathematics... then use whatever youre more comfortable with.

Radians only matter, mathematically, in higher level trig and calculus and beyond. There comes a point where you -have- to use radians... and must convert to them.

Answer 2

In radio astronomy, when we look at very small angles (fraction of seconds of arc) for objects that are very far away, the advantage is that
Sin(x) = x (and Tan(x) = x)
when x is in radian.

So an angle of 0.1" (0.1/3600 deg.) is
0.0000004848... radian.

If the objects is 2,400,000 light years away (e.g., M31: Andromeda galaxy), then the angle represents a distance of
0.0000004848 times 2,400,000 light years = 1.16 light-year (inside M31).

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When doing math and finding functions that are not readily apparent, radians allow you to turn trig functions into other types of functions (or vice versa).

Look at these functions:
SIn(x) x - (x^3)/3! + (x^5)/5! - (x^7)/7! ...
cos(x) = 1 - (x^2)/2! + (x^4)/4! - (x^6)/6! ...
when x is in radians.
and
e^x = (x^0)/0! + (x^1)/1! + (x^2)/2! + (x^3)^3! ...

There is just enough similarity that, under certain conditions, it is possible to go from non-trig. functions to trig. functions, and vice-versa (especially with complex numbers).

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When doing navigation (marine or air), we use degrees because all the information is normally found in degrees (latitudes and longitudes are given in degrees, so is azimuth -- sextants are marked in degrees and minutes).

However, there may be conventions to watch out for:

In math, angle zero is to the right (to the east, if you consider the graph like a map with north up), and the angle grows as you go counterclockwise.

In navigation, azimuth angles begin at North (towards the top of the map) and grow when gowing clockwise (090 = East, 180 = South, 270 = West --- 360 is North same as 000).

Great circle distances (in spherical trigonometry) are given in nautical miles which represent the length of an arc on Earth's surface, corresponding to an angle of 1' (= 1/60 deg.) at Earth's centre. That is why working with degrees is beneficial for navigation.

Answer 3

it depends on the application that your are using at the time.

for example, sometimes it is better to write fractions over a common denominator

(a + b)/c

and sometimes it is better to break them out

(a/c) + (b/c)

again it depnds on what you are doing.

However, learning radian measure is very beneficial to understanding and solving trig equations.