CHAPTER
TRIGONOMETRIC FUNCTIONS:
5
UNIT CIRCLE APPROACH
LEARNING OUTCOMES By the end of this chapter, you should be able to
Use reference number to find terminal point on the unit circle. Determine trigonometric function using properties of unit circle. Evaluate trigonometric function using definition of trigonometric functions. Sketch and transform graphs of sine, cosine, tangent, secant and cosecant. 5.1
UNIT CIRCLE This section is about properties of circle radius 1 unit and centred at the origin. The definition of unit circle is, Definition: Unit Circle The unit circle is the circle in
xy
plane centred at origin and has one unit
radius. The equation for unit circle is x2 y2 1
Example 1: Verify that the point
2 6 5 P , 7 7
is on the unit circle.
Solution We must verify that P satisfies equation of unit circle,
Since
and 2 6 x 7
5 y 7
, substitute into
x2 y2 1
x2 y2 1
Hence, 2
2 6 5 2 24 25 1 7 7 49 49
1
.
.
Example 2: Find
y
coordinate if the point
5 P , y 6
is on the unit circle in
Quadrant IV. Solution Since point P is on unit circle, substitute
into equation of unit circle 5 x 6
x2 y2 1
.
2
5 2 6 y 1 25 11 36 36 11 y 6 y2 1
Since the point is in Quadrant IV,
y
coordinate must be negative, thus
.
y
11 6
Terminal Points on The Unit Circle 2
Terminal point is a point of
measured in radians.
t
P ( x, y )
determined by the real number
which is
t
is the length of the arc intercepted by the angle,
in
radians. The terminal point
P ( x, y )
usually obtained when we travel start from point
and move along the unit circle in the direction of counter clockwise
clockwise
P ( x, y )
t 0
1,0
t 0
or
. Figure (a) and (b) show the relationship of terminal point
and . t
Terminal point
P(1,0)
counter clockwise
when travel
t 0
.
Figure (a) Terminal point
P(1,0)
counter clockwise
when travel
t 0 2
.
Figure (b) Note: Different values of
t
can determine the same terminal point.
3
Example 3: Find the terminal points on the unit circle determined by given real number . t (a)
(b)
2
2
(c)
(d)
3 2
3
Solution From the graph, (a) terminal point is
1,0
(b) terminal point is
1,0
(c) terminal point is
(d) 4
0,1
terminal point
1,0
We can configure other terminal points using
Figure (c)
t 4
.
P lies on line
yx
.
Since P is the intersection of line yx
and unit circle
x2 y2 1
x2 x2 1 2x 2 1 1 x2 2 1 2 x 2 2 y
2 2
P is in Quadrant I, thus terminal point
5
2 2 P , 2 2
.
,
We can use same method to find other terminal points. Table 1 shows the special values of . t
t
Terminal point
0
1,0
6
3 1 2 ,2
4
2 2 2 , 2
3
1 3 , 2 2
2
0,1 Table 1
The Reference Number Definition: Reference Number The reference number denoted as
associated with a real number, t is the
t shortest distance along the unit circle between terminal point and the
x
-axis.
Let’s observe this graph. We can determine the location of terminal point based on real number t
The reference number, is in
t between terminal point and axis
x Steps to find terminal point
P ( a,b)
6
using reference number,
1) Find reference number, . t 2) Find terminal point determined by . t 3) Choose the sign of terminal point
P ( a,b)
according to the quadrant in
which terminal point, lies. t
Example 4: Find the terminal points of given real number, t. (a)
(b)
(c)
7 6
5 4
(d)
4 3
41 4
Solution (a) The reference number,
.
The terminal point,
Terminal point of
negative and
y
(from Table 1).
is located at Quadrant III, where
is negative.
Therefore, terminal point of
(b) The reference number,
is
.
7
.
is
The terminal point,
(from Table 1).
Terminal point of
negative and
y
is located at Quadrant II, where
is positive.
Therefore, terminal point of
(c) The reference number,
is
Terminal point of
y
.
.
The terminal point,
negative and
is
(from Table 1).
is located at Quadrant III, where
is
is negative.
Therefore, terminal point of
(d) The reference number,
is
.
The terminal point,
Terminal point of
.
(from Table 1).
is located at Quadrant IV, where
8
is
positive
and
y
is negative.
Therefore, terminal point of
is
.
EXERCISE 1.1 1. (a) In your own words, explain what is unit circle. (b) State the equation of unit circle. 2. (a) What is reference number, ? t (b) Let the reference number be
t 3
for terminal point
P x, y
lies in Quadrant IV. 3. Verify the given point whether it is on the unit circle. (a) 5 11 , 6 6 4. Given the real number, t. Find, (i) reference number for each t. (ii) terminal point of for each t. P x, y (a)
3 4
(b)
7 6
(c)
(d)
11 3
9
31 6
(e)
29 6
, find P if it
5.2
TRIGONOMETRIC FUNCTIONS OF REAL NUMBERS In this section, we will see how the properties of unit circle can be used to define the trigonometric functions. Before that, what are trigonometric functions? Definition: Trigonometric Functions Let be the terminal point on the unit circle which determined by any P ( x, y ) real number,
t
and we define sin t y
csc t
cos t x
tan t
1 y denominator
1 sec t x
y x
cot t
0.WHY?
x y
Let’s create table of special values of the trigonometric functions using above definition and terminal points in Figure 1.
t
sin t
cos t
tan t
csc t
sec t
cot t
0
0
1
0
-
1
-
6
1 2
3 2
3 3
2
2 3 3
3
4
2 2
2 2
1
2
2
1
3
3 2
1 2
3
2 3 3
2
3 3
2
1
0
-
1
-
0
y
1
3 ; , 3 2 2
2 2 ; , 4 2 2
3 1 ; , 6 2 2
x
Figure 1
Table 1 Example 1: Find all six trigonometric functions determined by real number, . t
10
a)
b)
t 4
t
6
Solution a) Terminal point of
t 4
is
2 2 2 , 2
.
Then all six trigonometric functions are, sin t y csc t
1 y
cot t
x y
2 2 1
cos t x
2/2 2/2 2/2
b) Terminal point of
2 2
2 2
y 2/2 1 x 2/2 1 1 2 sec t 2 x 2/2 2 tan t
2
1
t 6
is
3 1 2 , 2
.
Then all six trigonometric functions are, sin t y
1 2
cos t x
csc t
1 1 2 y 1/ 2
cot t
x 3/2 3 y 1/ 2
3 2
tan t
y 1/ 2 1 3 x 3 3/2 3
sec t
1 1 2 2 3 x 3 3/2 3
Steps to evaluate the trigonometric functions for any real number . t 1. Find the reference number, . t 2. Determine the sign of the trigonometric function of
t
by locating which
quadrant that terminal point lies. Quadrant I
: All functions are positive
Quadrant II Quadrant III
: All functions are negative except for sine 11 : All functions are negative except for tangent
Quadrant IV
: All functions are negative except for cosine
Sine
All
Tangent
Cosine
3. Evaluate trigonometric function using reference number and add (obtained in step 2). Example 2: Evaluate the trigonometric functions. a) b) c) 5 7 5 sin cos tan 4 6 3 d)
sec 3
e)
d)
2 csc 3
25 cot 2
Solution a) Reference number,
For real number,
The sign for
.
the terminal point lies on Quadrant III.
is negative.
Evaluate:
b) Reference number,
For real number,
.
.
the terminal point lies on Quadrant II.
12
sign
The sign for
is negative.
Evaluate:
c) Reference number,
For real number,
The sign for
.
.
the terminal point lies on Quadrant IV.
is positive.
Evaluate:
d) Reference number,
For real number,
The sign for
.
.
the terminal point lies on Quadrant IV.
is the reciprocal of cosine. Cosine is negative in
Quadrant IV. Evaluate:
.
13
e) Reference number,
t 3
For real number,
The sign for
.
the terminal point lies on Quadrant II.
2 csc 3
is the reciprocal of sine. Sine is positive in
Quadrant II. Evaluate:
2 3 2 2 3 csc csc 3 3 3 3
f) Reference number,
For real number,
Evaluate
t 2
25 t 2
.
.
the terminal point is
25 x 0 cot 0 2 y 1
0,1
.
.
Even and Odd Properties. Let think of
the relationship of the
trigonometric
functions of
t
and
t
. We can
see that,
sin( t ) y sin t cos( t ) the x even-odd cos t From this, we can complete properties for other functions. Properties : Even and Odd Trigonometric Functions 14
Odd Functions sin( t ) sin t
Even Functions cos( t ) cos t
tan( t ) tan t
sec( t ) sec t
csc( t ) csc t cot( t ) cot t Example 3: Use even-odd properties to find the value of the trigonometric functions. a) sin 2
b)
c)
cos 4
csc 6
d)
sec 3
Solution a) Sine is an odd function. Therefore, 3 3 sin sin (1) 1 2 2 b) Cosine is an even function. Therefore, 2 cos cos 2 4 4 c) Cosecant is an odd function. Therefore, 1 1 csc csc 2 sin / 6 1/ 2 6 6 d) Secant is an even function. Therefore, 1 1 sec sec 2 3 3 cos / 3 1 / 2 Fundamental identities We use fundamental identities to relate any trigonometric function to any other. If we know the value of any trigonometric functions at , then we can use it to t find other trigonometric functions at . t
15
The fundamental identities are as follows. Fundamental Identities Phytagorean Identities
Reciprocal Identities
sin 2 t cos 2 t 1
csc t
1 sin t
tan 2 t 1 sec 2 t
sec t
1 cos t
1 cot 2 t csc 2 t
tan t
sin t cos t
cot t
1 cos t tan t sin t
Example 4: Evaluate all trigonometric functions of given. a) tan t 2
/2t
b)
terminal point of
4 csc t 3
c) sin t d)
1 3
12 cos t 13
t
t
by using the information
is in Quadrant I
sec t 0 terminal point of
t
is in Quadrant IV
Solution a) Using phtagorean identity
.
2 2 1 sec 2 t sec 2 t 5 sec t 5 1 5 cos t 1 cos t 5 16
Since
t
is in Quadrant II, cosine is negative. Similar with secant.
sec t 5 , cos t
1 5
Using phytagorean identity
sin 2 t cos 2 t 1
2
1 sin t 1 5 1 4 sin 2 t 1 5 5 4 2 5 2 5 sin t 5 5 5 5 2
Using reciprocal identity
csc t
cot t
1 2/ 5
2/ 5
and cot t
cos t sin t
5 2
1/ 5
1 csc t sin t
1 5
5 1 2 2
Answers: tan t 2
,
, , 1 5 sec t 5 , cos t cot t sin t 2 5 csc t 2 5 2 5 1
,
Trigonometric Graphs In this section, you will learn how to sketch the graph of sine, cosine, tangent, cotangent, secant and cosecant. Graph
Properties
Graph
Name
(a=1,k=1)
17
Equation: Sine
y a sin kx Period:
2 / k
Amplitude:
a
Equation: Cosine
y a cos kx Period:
2 / k
Amplitude:
a
Equation: Tangent
y a tan kx Period:
/k
Equation: Cotangent
y a cot kx Period:
Cosecant
/k
Equation: y a csc kx Period:
2 / k
18
Equation: Secant
y a sec kx Period:
2 / k
Notes:
k
is a constant and
k 0
Transformation of Graphs The focus in this section is what happened to the sine, cosine and tangent curve if we shift them to the right or to the left. Let’s see some properties that will helps us to sketch those curves. Graph’s
Properties
Example of Shifted Graph
Name Shift to the right
/6 Equation: Sine Period:
Period: 2π
y a sin k ( x b)
2 / k
Amplitude:
Phase shift: Interval:
Amplitude=2 y 2 sin x
a
b
b, b 2 / k
19
6
Equation: Cosine
Period:
y a cos k ( x b)
2 / k
Amplitude:
Phase shift: Interval:
y 5 cos 3x 7
b
b, b 2 / k
Equation: Tangent Period:
a
y a tan k ( x b)
/k
Phase shift:
b y tan x / 4
Equation: Cotangent Period:
y a cot k ( x b)
/k
Phase shift:
b y 5 cot 3 x / 2
EXERCISE 5.2
20
21