IN THE RADIAN SYSTEM of angular measurement, the measure of one full circle is 2π.
(In the next Topic, Arc Length, we will see the actual definition of radian measure.)
Half a circle, then, is π. And, most important, each right angle is half
Radians into degrees
The following radian measures come up frequently, and the student should know their degree equivalents:
is half of , a right angle, and so it is equal to 45°.
Problem 1 . Convert each of these radian measures into degrees.
The student should know these.
To see the answer, pass your mouse over the colored area.
Problem 2. Convert each of these radian measures into degrees.
Degrees into radians
360° = 2π.
When we write 2π, we mean 2π radians, which is approximately 6.28 radians. However, we normally omit the word radians. As we will see in the next Topic, Arc length, the radian measure can be any real number.
Problem 3. The student should begin by knowing these.
Example 1. Convert 120° into radians.
Solution. We can go from what we know to what we do not know. In the most important cases we can recognize the number of degrees as a multiple of 90°, or 45°, or 60°, or 30°; or as a part of 360°.
Or, since 120° is a third of 360°, which is 2π, then
Problem 4. Convert each of the following into radians.
72° is thus a fifth of a revolution.
As a last resort, proportionally,
Example 3. Change 140° to radians.
Angles are coterminal if they have the same terminal side.
θ is coterminal with −φ. They have the same terminal side.
θ + φ = 2π,
θ = 2π − φ . . . . (1)
Example 4. Name in radians the non-negative angle that is coterminal
Answer. Let us call that angle θ. Then according to line (1),
Alternatively, since the numerator is 2π, then is a fifth of a revolution in the negative direction.
Therefore θ will be four fifths of a revolution in the positive direction.
Problem 5. Name in radians the non-negative angle that is coterminal with each of the following, and is less than 2π.
This is a third of a revolution in the negative direction. Therefore, θ will be two thirds of a revolution in the positive direction.
The multiples of π
Starting at 0, let us go around the circle a half-circle at a time. We will then have the following sequence, which are the multiples of π:
0, π, 2π, 3π, 4π, 5π, etc.
The point to see is that the odd multiples of π --
π, 3π, 5π, 7π, etc.
-- are coterminal with π. While the even multiples of π --
2π, 4π, 6π, etc.
-- are coterminal with 0.
If we go around in the negative direction,
we can make a similar observation.
Problem 6. Name in radians the non-negative angle that is coterminal with each of the following, and is less than 2π.
a) -π π b) -2π 0 c) -3π π
d) -4π 0 e) -5π π
f) 3π π g) 4π 0 h) 5π π
i) 6π 0 j) 7π π
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