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The Sine and Cosine Functions
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The Functions Sine and Cosine
In this section we make the transition from thinking of cos! and sin ! as trigonometric ratios of
the sides of a right triangle to being functions of ! . Moreover, we show how by using the
concept of reference angle and by accounting for the sign of these functions on the different
quadrants of the coordinate system, one can evaluate the value of cos! and sin ! of any angle !
from known values of cosine and sine on the first quadrant.
We saw previously that any central angle " is related to exactly one point (x, y) on the unit
circle, and that in such case, (x, y) is called the corresponding point of " . For example,
( 3 / 2, 1 / 2) is the corresponding point of ! = 30!. We also saw that the coordinates of a point
(x, y) on the unit circle give the values of cos " and sin" , respectively. For example,
cos 30! = 3 2 and sin 30! = 1 2 .
It is also true, that many angles can have the same corresponding point. For example, all angles
!
separated from " = 30 by 360! or a multiple of 360! , in the clockwise or counterclockwise
direction, have the same corresponding point ( 3 / 2, 1 / 2) , and thus have the same value of sine
and cosine. For example, cos 390! = 3 / 2 and cos! 330! = 3 / 2 , while sin 390! = 1 / 2 and
sin! 330! = 1 / 2 .
All these facts point to the fact that there exist a many-to one relation between ! and cos! , and
likewise between ! and sin ! . In most College Algebra courses one learns that a many-to-one
relation is a type of function. Therefore, we have that the value of cos! is a many-to-one
function of " and we write f (θ ) = cos θ , and similarly for sin ! we write f (θ ) = sin θ . These
are called cosine and sine functions of θ . These functions can be easily used for finding the
value of the other four trigonometric functions, namely f (! ) = tan ! , f (! ) = cot ! , f (! ) = sec! ,
and f (! ) = csc! by recalling that all of them can be written in terms of cos! and sin !
according to (7).
!
!
The x-axis and the y-axis of the coordinate system divide the plane into four quadrants. If a unit
circle (of radius 1) is centered at the origin of this system as shown in Figure 10, the axes divide
it into four equal parts. We say that an angle belongs to a particular quadrant when its terminal
side is on that quadrant. Thus the 1/4 of this circle that lies above the positive x-axis contains the
The Sine and Cosine Functions
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points that correspond to angles 0! < ! < 90! and they belong to quadrant I (QI) of the coordinate
system because their terminal side is on this quadrant. Similarly, the second quadrant (QII)
contains angles 90! < ! < 180! , the third quadrant (QIII) contains 180! < ! < 270! , while the
fourth quadrant (QIV) contains 270! < ! < 360! .
QII
QI
QIII
QIV
Figure 10. A unit circle and the quadrants of a rectangular coordinate system.
Example 15. Determine the quadrant of angle 45! .
Solution:
This angle has a terminal on QI, therefore 45! belongs to QI.
Example 16. Determine the quadrant of 281! .
Solution:
This angle is a bit larger than 270! but less than 360! . Therefore this angle has a terminal side in
QIV, and therefore 281! belongs to QIV.
Angles that are larger than 360! or that are negative angles can also have their terminal side on
one of the four quadrants.
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Example 17. Determine the quadrant of angle 756! .
Solution:
Angle 756! is larger than 720! by 36! . Note that the angle 720! corresponds to two complete
turns around the unit circle. This means that angle 756! has its terminal side on QI, and therefore
756! belongs to QI.
Example 18. Determine the quadrant of !5" / 6 .
Solution:
To find the terminal side of angle !5" / 6 we must go clockwise 5! / 6 units from the positive
x-axis. Angle !5" / 6 is ! / 6 units short of angle !" . The terminal side of !5" / 6 is on QIII,
and therefore !5" / 6 belongs to QIII.
The sign of all six trigonometric functions on the four quadrants can be easily remembered by
the sentence “All Students Take Calculus.” The letter A of the first word stands for the fact that
all six trigonometric functions are positive QI. In Figure 11 we illustrate this fact by placing an
A on QI followed by a + sign.
QII
QIII
QI
S (+)
A (+)
T (+)
C (+)
QIV
Figure 11. The sign of all trigonometric functions is shown for the four quadrants.
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Similarly, the letter S (of the second word) stands for the fact that only Sine and cosecant (its
reciprocal) are positive on QII while the rest of the functions are negative; T stands for the fact
that only Tangent and cotangent are positive on QIII while the rest are negative; and C stands for
the fact that Cosine and secant are the only positive functions on QIV.
" 5! %
" 5! %
" 5! %
Example 19. Determine (a) the sign of cos $ ' , (b) sin $
, and (c) tan $
.
'
# 6 &
# 6 &
# 6 '&
Solution:
(a) The cosine function according to Figure 11 is positive on QI and QIV. The angle 5! / 6
" 5! %
belongs to QII, and therefore, the value of cos $ ' is negative.
# 6 &
(b) The sine function is positive on QI and QII. The angle 5! / 6 belongs to QII. Therefore,
" 5! %
the value of sin $
is positive.
# 6 '&
" 5! %
(c) The value of tan $
is negative because by (7) tangent is the quotient of sine and
# 6 '&
cosine. From (b) we know that the sine is positive and from (a) that cosine is negative.
" 5! %
Therefore, tan $
is negative. This agrees with Figure 11 that shows that tangent is
# 6 '&
positive only on QI and QIII.
In order to be able to determine the value of a trigonometric function for any angle (without a
calculator) from known values of cosine and sine on QI, we need the concept of a reference
angle.
Let us assume that ! represents an angle on QII, QIII, or QIV. The reference angle of ! is
always measured with respect to the part of the x-axis that is closest to the terminal side of ! . If
! has its terminal side on QII or QIII, then the reference angle of ! is measure with respect to
the negative side of the x-axis, while if ! is on QIV it is measured with respect to the positive xaxis. Note that the negative side of the x-axis corresponds to angle 180! , while the positive side
of the x-axis to 360! .
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We find the reference angle of an angle ! that is less than 360! by finding the difference
between ! and 180! or ! and 360! . In order to get the right sign for the reference angle we
always subtract the smaller angle from the bigger one of the two.
Example 20. Find the reference angle of ! = 5" / 6 .
Solution:
This angle has a terminal side on QII, then its reference angle is the difference between ! (the
angle corresponding to the negative x-axis) and 5! / 6 . Note that ! is the bigger than 5! / 6 ,
and therefore we get:
Reference Angle = ! "
5! !
= .
6
6
Example 21. Find the reference angle of ! = 210! .
Solution:
This angle has a terminal side on QIII. The part of the x-axis closest to it corresponds to 180! .
The reference angle is the difference between 210! and 180! , and we get:
Reference Angle = 210! ! 180! = 30!.
Example 22. Find the reference angle of ! = 11" / 6 .
Solution:
This angle has its terminal side on QIV, is its reference angle is calculated as follows:
Reference Angle = 2! "
11! !
= .
6
6
In the previous three examples, the reference angle was determined to be ! / 6 (or 30! ) . There are
angles bigger than 2! (or 360! ) that also have ! / 6 as reference angle.
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Example 23. Find the reference angle of ! = 17" / 6 (or 510! ) .
Solution:
Note that this angle belongs to QII, and therefore its reference angle must be measured with
respect to the negative side of the x-axis. However, it is not the same situation as for Example 20
because this angle measures between one and two complete turns around the circle. In this case
we still measure the reference angle with respect to negative side of the x-axis but now it
corresponds to 3! (or 540! ) . Thus the reference angle is evaluated as follows:
Reference Angle = 3! "
17! !
= .
6
6
Example 24. Find the reference angle of ! = "5# / 6 (or -150! ) .
Solution:
The approach to finding the reference angle when ! is a negative angle, is similar to what we
have been doing so far. We still find the reference angle by evaluating the difference between !
and the negative side of the x-axis, but we must remember that the negative x-axis now
corresponds to !" (or -180! ) . The reference angle of ! = "5# / 6 (or -150! ) is evaluated as
Reference Angle = !
follows:
Note that !
5"
"
! (!" ) = .
6
6
5"
> !" .
6
The value of f (θ ) = cos θ for any angle ! , can be found using the value of cosine of the
reference angle of ! . The final answer must take into account of the sign of the function at the
quadrant where ! belongs.
" 5! %
Example 25. Find cos $ ' using the concept of reference angle and accounting for the sign of
# 6 &
the cosine function in the four quadrants.
Solution:
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Here ! = 5" / 6 (or 150! ) . The terminal side of ! is on QII and therefore it has ! / 6 as
reference angle (see Example 1). Also, the sign of f (θ ) = cos θ is negative on QII according to
" 5! %
"!%
Figure 11. Therefore, cos $ ' is equal in magnitude to the value of cos $ ' but it is negative,
# 6 &
# 6&
since cosine is negative on QII.
" 3%
3
" 5! %
"!%
cos $
= ( cos $ ' = ( $
=(
.
'
'
# 6 &
# 6&
2
# 2 &
Therefore:
" 7! %
Example 26. Find sin $
.
# 4 '&
Solution:
Angle
7!
belongs to QIV, and the function sine is negative in this quadrant. Also, the reference
4
angle of angle
7!
!
2
2
"!%
" 7! %
=(
is , and sin $ ' =
. Therefore, sin $
.
'
# 4&
# 4 &
2
2
4
4
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The following table lists commonly used angles and their corresponding point on the unit circle.
It also includes the values of sine and cosine for these angles. You must make an effort to
become familiar with these values, and ideally you should memorize them.
Corresponding
! (degrees)
! (radians)
point (!, !)
cos !
sin !
0o
0
(1,0)
1
0
30o
π
6
( 3 / 2, 1 / 2)
3/2
45o
π
4
( 2 2, 2 2)
2 /2
60o
π
3
(1 2, 3 2)
1/ 2
90o
π
2
(0,1)
0
1
180o
π
(−1,0)
!1
0
270o
3π
2
(0, −1)
0
!1
360!
2π
(1,0)
1
0
1/ 2
2 /2
3/2
Table 4. List of commonly used angles, their corresponding point on the unit circle, and their
corresponding values of cosine and sine.
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It is customary to use the letter x instead of " and to write f (x) = sin x and f (x) = cos x . When
one looks at the graphs of these functions, the independent variable x is read on the x-axis, while
the value of the function f ( x) (dependent variable) is read on the y-axis. Figure 13 shows the
graphs of sin x and cos x from 0 radians to 2" radians.
!
Figure 13. Illustration of a unit circle and the graphs of sin x (on top) and cos x (on bottom).
Notice that both functions have a range that goes from –1 to 1, and both are many-to one
functions (by the Horizontal Line Test).
In Section 1.8 of Adler’s textbook, you will study how the transformations of the trigonometric
functions f (x) = sin x , f (x) = cos x , are used in the modeling the oscillatory behavior of
biological systems.