Download MAT 1275: Introduction to Mathematical Analysis Dr

MAT 1275: Introduction to
Mathematical Analysis
Dr. A. Rozenblyum
V. Basic Concepts of Trigonometry
D. Trigonometric Functions of Any Angle
Trigonometric Functions of Any Angle
In section V.B, we defined trig functions for arbitrary angles. Here we will develop a
technique on how to actually calculate trig functions for given angles. The basic tools that
we will use are coterminal angles and reference angles.
As we mentioned in section V.A, one “geometric” angle (two rays, coming from the
same point) corresponds to infinite number of “trigonometric” angles (with additional
information about direction of rotation). All such angles are called coterminal. In the
following definition, we consider angles in standard position (the initial side coincides
with the positive part of x-axis, and the vertex is in the origin).
Definition. Two angles in standard positions are called coterminal, if their terminal sides
coincide.
If  is any angle, then all angles coterminal to  can be described by the formula
  360  n , where n is any integer. In other words, any two coterminal angles differ

from each other by a multiple of 360 (which represents a full rotation around circle).
Obviously that trig functions of coterminal angles are equal. For a given angle, we can


always find a unique coterminal angle that is in the range [0 , 360 ) . This coterminal
angle is the (positive) remainder when we divide given angle by 360.
Example 1. Find positive angles that are less than 360 and coterminal with

1) 870 .

2)  855 .
Solution. We need to find positive remainders when we divide given angles by 360.

1) 870  360  2  150 , so the remainder is 150. The coterminal angle is of150 .
2)  855  360  3  225 , so the remainder is 225. The coterminal angle is of
225 .
Trigonometric Functions of Any Angle
Now consider the concept of reference angle. This is a useful tool to reduce calculation of
trig functions of arbitrary angles to acute angles.
Definition. Let  be an arbitrary angle in standard position. Angle   is called the
reference angle to  , if it satisfies three conditions:
1) Angle   is an acute angle.
2) Terminal side of   coincides with the terminal side of  .
3) Initial side of   coincides with the positive or negative parts of the x-axis.
The position of the initial side of reference angle   depends on in what quadrant angle
 is located. Bellow, we consider all four cases. (We may assume that angle  is a

positive and is less than 360 ; otherwise, we can replace  with the corresponding
coterminal angle).
Trigonometric Functions of Any Angle
The position of the initial side of reference angle   depends on in what quadrant angle
 is located. Bellow, we consider all four cases. (We may assume that angle  is a

positive and is less than 360 ; otherwise, we can replace  with the corresponding
coterminal angle).
1) Angle  is located in the first quadrant. In this case   coincides with  .
2) Angle  is located in the second quadrant. In this case    180   :
3) Angle  is located in the third quadrant. In this case      180 :
4) Angle  is located in the fourth quadrant. In this case    360   :
Trigonometric Functions of Any Angle
Reference angle is a useful tool to find the absolute value of trig functions of arbitrary
angles: for any angle  , the absolute value of any trig function of  is equal to the same
trig function of the reference angle   .
We can use the following steps to calculate a trig function of given angle  :
1) If necessary, replace angle  with the coterminal angle in the interval from0 to
360 .
2) Identify in which quadrant angle  is located.
3) Determine the sign of the trig function in question.
4) Find the reference angle   .
5) Calculate trig function of   . The result gives the absolute value of the trig
function of  .
6) Write final answers, based on the results in steps 3 and 5.
Trigonometric Functions of Any Angle
Example 2. Calculate cos870 .
Solution. We will follow the above steps.

1) In example 1, we found that corresponding coterminal angle is150 .
2) Angle 150 is located in the second quadrant.
3) In the second quadrant, cosine is negative. Therefore, cos870  0 .
4) Reference angle    180  150  30 .
3
.
2

5) cos   cos 30 

6) Final answer: cos870  

3
.
2


Example 3. Calculate sin  855 .
Solution.

1) In example 1, we found that corresponding coterminal angle is 225 .
2) Angle 225 is located in the third quadrant.


3) In the third quadrant, sine is negative. Therefore, sin  855  0 .
4) Reference angle    225  180  45 .
5)
sin    sin 45 

2
.
2


6) Final answer: sin  855  
2
.
2
Trigonometric Functions of Any Angle
Example 4. Show that


2) cos  180    cos .
3) tan   180   tan  (periodic property of tangent).

1) sin   180   sin  .


Solution.
1) If terminal side of the angle  is in the first quadrant, then the terminal side of angle
  180 is in the third quadrant (and vice versa). If terminal side of the angle  is in

the second quadrant, then the terminal side of angle   180 is in the fourth quadrant

(and vice versa). Therefore, reference angles of  and   180 are equal (they are

vertical angles), so absolute values of sin   180 and sin  are the same. Also,
sin   180 and sin  have opposite signs.




2) The same reasoning as in 1).



3) tan   180 




sin   180
 sin sin


 tan  .

cos   180
 cos cos
End of the Topic