Download Chapter 5: Trigonometric Functions of Real Numbers

c
Math 150, Fall 2008, Benjamin
Aurispa
Chapter 5: Trigonometric Functions of Real Numbers
5.1 The Unit Circle
The unit circle is the circle of radius 1 centered at the origin. Its equation is
x2 + y 2 = 1
Example: The point P (x, 13 ) is on the unit circle in Quadrant II. Find x.
Given a real number t, suppose we want to find the point on the unit circle that is a distance t from the
point (1, 0) along the circumference of the circle. If t is positive we move in the counterclockwise direction.
If t is negative we move in the clockwise direction.
The point P (x, y) that is found is called the terminal point determined by t.
Remember that the circumference of the unit circle is 2π.
If t = π2 , we have traveled 14 of the circumference ( 41 of 2π is
π
2 ).
So the terminal point is
If t = π, we have traveled half of the circumference. So the terminal point is
If t =
3π
2 ,
we have traveled
3
4
of the circumference ( 34 of 2π is
3π
2 ).
.
.
So, the terminal point is
.
For other values of t, finding the terminal point is not always straightforward. The terminal points listed
below are the common ones that will be seen often and will be used to find terminal points for values of t
in other quadrants. These are the ones you should memorize.
Value of t
0
Terminal Point
π
6
π
4
π
3
π
2
π
3π
2
1
c
Math 150, Fall 2008, Benjamin
Aurispa
The reference number t associated with t is the shortest distance along the unit circle between the x-axis
and the terminal point determined by t.
If t is in Quadrants I or IV, then the reference number t is measured from the positive x-axis.
If t is in Quadrants II or III, then the reference number t is measured from the negative x-axis.
Note: The reference number is always between 0 and
π
2.
Examples: Find the reference numbers for the following values of t.
• t=
5π
6
• t = − 7π
4
• t=
11π
3
We use the reference number to find terminal points for values of t not in Quadrant I. Once we know the
reference number, we use the terminal point associated with this value and just adjust the signs of the
coordinates as needed.
Examples: Find the terminal point for the following values of t by first finding the reference number.
• t=
2π
3
• t=
11π
6
• t = − 19π
4
2
c
Math 150, Fall 2008, Benjamin
Aurispa
5.2 Trigonometric Functions of Real Numbers
Definition of Trig Functions:
Let t be any real number and let P (x, y) be the terminal point on the unit circle determined by t. We define
sin t = y
csc t =
1
sin t
=
1
y
(y 6= 0)
cos t = x
sec t =
1
cos t
=
1
x
(x 6= 0)
cot t =
1
tan t
=
cos t
sin t
tan t =
sin t
cos t
=
y
x
(x 6= 0)
=
x
y
(y 6= 0)
Fill in the table with the 6 trig values for each of the given values of t.
Value of t
0
sin t
cos t
tan t
csc t
sec t
cot t
π
6
π
4
π
3
π
2
π
3π
2
Different trig functions have different signs depending on the quadrant the terminal point is in.
If you need it, the following can help you remember which trig functions are positive in each quadrant:
All Students Take Calculus.
When the reference number for a value of t is in the table above, we can find the trig value of t by finding
the trig value of the reference number and changing the sign depending on the quadrant the terminal point
of t is in.
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c
Math 150, Fall 2008, Benjamin
Aurispa
Evaluate the following:
7π
• tan( 7π
6 ) and csc( 6 )
3π
• cos( 3π
4 ) and cot( 4 )
• sin(− π3 ) and sec(− π3 )
• tan( 8π
3 )
Find all trig values for a number t if the terminal point of t is
4
8
9,
√
17
9
.
c
Math 150, Fall 2008, Benjamin
Aurispa
For values of t that do not have the standard reference numbers and where you don’t know the terminal
point, you can evaluate on your calculator.
Evaluate csc 4.7 and cot 45
If sin t < 0 and cot t > 0 what quadrant is the terminal point of t in?
Find the sign of
tan t sec t
if the terminal point of t is in Quadrant IV.
sin2 t
Pythagorean Identities: (These are used a lot.)
sin2 t + cos2 t = 1
tan2 t + 1 = sec2 t
1 + cot2 t = csc2 t
Note: The last two identities follow from the first identity when you divide by either sin2 t or cos2 t.
Find the values of all the trig functions of t given that cos t = − 54 and that the terminal point of t is in
Quadrant III.
Write cos t in terms of sin t given that the terminal point of t is in Quadrant IV.
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c
Math 150, Fall 2008, Benjamin
Aurispa
Write tan t in terms of cos t given that the terminal point of t is in Quadrant II.
Even-Odd Properties
Sine and Tangent (and their reciprocals) are ODD functions. So sin(−t) = − sin t and tan(−t) = − tan t.
Cosine (and its reciprocal) are EVEN functions. So cos(−t) = cos t.
Find cos(− π6 ) and sin(− π4 ).
Determine whether f (x) = tan x cos x is odd, even, or neither.
Domains of Trig Functions
sin and cos are defined for all real numbers.
tan and sec are defined for all real numbers other than
π
2
+ nπ for any integer n. Why?
cot and csc are defined for all real numbers other than nπ for any integer n. Why?
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