Trigonometry

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14

RADIAN MEASURE

Radians into degrees

Degrees into radians

Coterminal angles

The multiples of π


IN THE RADIAN SYSTEM of angular measurement, the measure of one revolution is 2π.

(In the next Topic, Arc Length, we will see the actual definition of radian measure.)

Radian measure

Half a circle, then, is π.  And, most important, each right angle is half

  of π:   π
2		 .
  Three right angles will be  3·   π
2		   =  3π
 2 		 .
  Five right angles will be   5π
 2 		 .  And so on.

Radians into degrees

The student should have a clear picture of the following:

Radian measure

(Topic 6 and Topic 7.)

π
4		   is half of   π
2		  , a right angle, and so it is equal to 45°.   (Skill in

Arithmetic, Multiplying and dividing fractions, Question 5.)

Equivalently,  π
4		   is of one quarter of π.
π
3		  is a third of π, and so is equal to 180° ÷ 3 = 60°.
π
6		  is a sixth of π, and so is equal to 180° ÷ 6 = 30°.
5π
 4 		   =  5·  π
4 		   =  5· 45° = 225°.
2π
 3 		   is a third of 2π.   A third of a revolution = 360° ÷ 3 = 120°.

Problem 1.   Convert each of these radian measures into degrees.

Problem 1.     The student should know these.

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   a)   π   180°        b)    π
2		    90°        c)    π
3		    60°        d)    π
6		    30°        e)    π
4		    45°

Problem 2.   Convert each of these radian measures into degrees.

   a)      π
8		     22½°.   π
8		  is half of  π
4		 .
   b)    2π
 5		     72°.   2π
 5 		 is a fifth of 2π  , which is a fifth of a 360°.
   c)    7π
 4		    = 7·  π
4 		 = 7· 45° = 315°
   d)     9π
 2		    = 9·  π
2 		 = 9· 90° = 810°
   e)     4π
 3		    = 4·  π
3 		 = 4· 60° = 240°
   f)     5π
 6		    = 5·  π
6 		 = 5· 30° = 150°
   g)    7π
 9		  

Problem 3.   Evaluate the following.  (See Topic 6 and Topic 7.)

  a)  cos  π
6		  = 
 2		     b)  sin  π
6		  =  1
2		     c)  tan  π
4		  =  1
 
  d)  cot  π
3		  =   1 
     e)  csc  π
6		  =  2      f)  sec  π
4		  = 

Problem 4.   In terms of radians, what angle is the complement of an

  angle θ ?    π
2		  −  θ

Problem 5.   A function of any angle is equal to the cofunction of its complement. (Topic 5.)  Therefore, in terms of cofunctions:

  a)  sin θ =  cos ( π
2		  −  θ )     b)  cot θ =  tan ( π
2		  −  θ )
  c)  sec ( π
2		  − θ)  = csc θ

Degrees into radians

360°  =  2π.     180°  =  π.

When we write, 180° = π, we mean that it equals π radians, which is approximately 3.14 radians.  However, we normally omit the word radians.  For, as we will see in the next Topic, Arc length, the radian measure can be any number.

Example 1.   Convert 120° into radians.

Solution.   We can go from what we know to what we don't know.  Since

  60° =  π
3		 , then
120° = 2· 60° =  = 2·  π
3		  =  2π
 3		 .

Or, since 120° is a third of 360°, which is 2π, then

120°  =   2π
 3		 .
   Example 2.   225° = 180° + 45°  =  π +  π
4		   =   5π
 4

In general, proportionally,

so that

Example 3.   Change 140° to radians.

  Solution.     140
180		 · π   =   7
9		 · π   =   7π
 9		 ,

upon dividing both the numerator and denominator first by 10 and then by 2.  (Lesson 21 of Arithmetic, and Lesson 1.)

Problem 6.   Change each of the following into radians.

   a)  0°   0 radians       b)  180°   π       c)  90°   π
2		       d)  45°   π
4		       e)  270°   3π
 2
   f)  60°   π
3		               g)  30°   π
6		       h)  720°   = 2· 360° = 2· 2π = 4π
   i)  210°   = 7· 30° = 7·  π
6		  =  7π
 6
   j)  300°   = 5· 60° = 5·  π
3		  =  5π
 3
   k)  135°   = 90° + 45° =  π
2		  +  π
4		  =  3π
 4
   l)  72°  =   72
180		 · π =  2
5		 · π =  2π
 5

Coterminal angles

Angles are coterminal if they have the same terminal side.

Coterminal angles

θ is coterminal with −φ.  They have the same terminal side.

Notice that

θ + φ  =  2π,

so that

θ  =  2π − φ .  .  .  .  .  .  .  . (1)

Example 4.   Name in radians the non-negative angle that is coterminal

   with − 2π
 5		 , and is less than 2π. 

Answer.  Let us call that angle θ.  Then according to line (1),

θ  =  2π −  2π
 5		   =   10π − 2π
        5		   =   8π
 5

See Lesson 23 of Algebra, Example 7.

Problem 7.   Name in radians the non-negative angle that is coterminal with each of the following, and is less than 2π.

   a)  − π
6		       θ  =  2π −  π
6		   =   12π − π
     6		   =   11π
  6
   b)  − 3π
 4		     θ  =  2π −  3π
 4		   =   8π − 3π
     4		   =   5π
 4
   c)  − 4π
 3		     θ  =  2π −  4π
 3		   =   6π − 4π
     3		   =   2π
 3

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 multiples of pi

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,

The multiples of pi

we can make a similar observation.

Problem 8.   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 



Next Topic:  Arc Length


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