We can find
coterminal angles measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.
Given an angle greater than
find a coterminal angle between 0 and
Subtract
from the given angle.
If the result is still greater than
subtract
again until the result is between
and
The resulting angle is coterminal with the original angle.
Finding coterminal angles using radians
Find an angle
that is coterminal with
where
When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of
radians:
The angle
is coterminal, but not less than
so we subtract another rotation.
Recall that the radian measure
of an angle was defined as the ratio of the
arc length
of a circular arc to the radius
of the circle,
From this relationship, we can find arc length along a circle, given an angle.
Arc length on a circle
In a circle of radius
r , the length of an arc
subtended by an angle with measure
in radians, shown in
[link] , is
Given a circle of radius
calculate the length
of the arc subtended by a given angle of measure
If necessary, convert
to radians.
Multiply the radius
Finding the length of an arc
Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.
In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?
Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day.
Let’s begin by finding the circumference of Mercury’s orbit.
Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled.
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