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445.102 Mathematics 2
Module 4
Cyclic Functions
Lecture 2
Reciprocal Relationships
445.102
Lecture 4/2
Administration
Last
Lecture
Looking Again at the Unit Circle
Some Other Functions
Equations with Many Solutions
Summary
Administration
Chinese
Tutorials
Text Handouts
Modules 0, 1, 2
—> p52
Module 3
—> pp87 - 109
Module 4
—> pp77 - 88
This Week’s Tutorial
Assignment 4 & Working Together
445.102
Lecture 4/2
Administration
Last
Lecture
Looking Again
at the Unit Circle
Some Other Functions
Equations with Many Solutions
Summary
Radians
A mathematical measure of angle is
defined using the radius of a circle.
1 radian
sin(ø)
1
ø
sin(ø)
Post-Lecture Exercise
1
45° = π/4 radians
60° = π/3 radians
80° = 4π/9 radians
2 full turns = 4π radians
270° = 3π/2 radians
2
3
π radians = 180°
3 radians = 171.9°
6π radians = 3 turns
f(x) = sin x is an ODD function.
4
f(2.5) = 0.598
5
6
f(20) = 0.913
f(–4) = 0.757
f–1(0.5) = 0.524 f–1(0.3) = 0.305 f–1(–0.6) = –0.644
The domain of f(x) = sin x is the Real Numbers
The domain of the inverse function is –1 ≤ x ≤ 1
f(π/4) = 0.707
Lecture 4/1 – Summary
There
are many functions where the
variable can be regarded as an ANGLE.
 One way of measuring an angle is that
derived from the radius of the circle. This is
called RADIAN measure.
From the UNIT CIRCLE, we can see that
the SINE of an angle is the height of a
triangle drawn inside the circle. Sine(ø)
then becomes a function depending on the
size of the angle ø.
The Sine Function
(Many Rotations)
1.00
f(ø) = sin ø
0.50
-2š
-š
š
-0.50
-1.00
2š
3š
4š
Preliminary Exercise
1.00
f(ø) = sin ø
0.50
-2š
-š
š
-0.50
-1.00
2š
3š
4š
445.102
Lecture 4/2
Administration
Last
Lecture
Looking Again
Some
at the Unit Circle
Other Functions
Equations with Many Solutions
Summary
C(ø)
1
ø
C(ø)
cos(ø)
1
ø
cos(ø)
tan(ø)
tan(ø)
ø
1
Constructions on the Unit Circle
1
tan(ø)
sin(ø)
ø
cos(ø)
The Cosine Function
(Many Rotations)
1.00
f(ø) = cos ø
0.50
-2š
-š
š
-0.50
-1.00
2š
3š
4š
The Tangent Function
(Many Rotations)
1.00
f(ø) = tan ø
0.50
-2š
-š
š
-0.50
-1.00
2š
3š
4š
445.102
Lecture 4/2
Administration
Last
Lecture
Looking Again at the Unit Circle
Some
Other Functions
Equations
Summary
with Many Solutions
The Secant Function
secant
sec ø/ = sec ø = 1/
1
cos ø
sec ø
1
cos(ø)
1
Inverse Functions
The sine function maps an angle to a
number.
e.g. sin π/4 =0.707
 The inverse sine function maps a number to
an angle.
e.g. sin-10.707 = π/4
 Note the difference between:

The inverse sine: sin-10.707 = π/4
The reciprocal of sine:
(sin π/4)-1 = 1/(sin π/4) = 1/0.707 = 1.414
Inverse Functions
Here is a quick exercise..........
 (remember to give your answers in radians):

1.
 2.
 3.
 4.
 5.
 6.

What angle has a sine of 0.25 ?
What angle has a tangent of 3.5 ?
What angle has a cosine of –0.4 ?
What is sec π/2 ?
What is cot 5π/3 ?
What is arctan 10 ?
445.102
Lecture 4/2
Administration
Last
Lecture
Looking Again at the Unit Circle
Some Other Functions
Equations
Summary
with Many Solutions
An Equation
2cos ø – 0.6 = 0
2cos ø = 0.6
cos ø = 0.3
1.00
f(ø) = cos ø
0.50
-2š
-š
š
-0.50
-1.00
2š
3š
4š
An Example ....
4sin ø + 3 = 1
4sin ø = –2
sin ø = –0.5
ø = sin -1(–0.5) = –0.524
–0.524, π+0.524, 2π–0.524, 3π+0.524,....
nπ+0.524 (n = 1,3,5,7,....)
nπ–0.524 (n = 0,2,4,6,....)
nπ+0.524 (n = ...-5,-3,-1,1,3,5,7,....)
nπ–0.524 (n = ...-6,-4,-2,0,2,4,6,....)
An Example ....
4sin ø + 3 = 1
4sin ø = –2
sin ø = –0.5
ø = sin -1(–0.5) = –0.524
–0.524, π+0.524, 2π–0.524, 3π+0.524,....
nπ+0.524 (n = 1,3,5,7,....)
nπ–0.524 (n = 0,2,4,6,....)
nπ+0.524 (n = ...-5,-3,-1,1,3,5,7,....)
nπ–0.524 (n = ...-6,-4,-2,0,2,4,6,....)
A Special Triangle
1 unit
1 unit
A Special Triangle
1
1
A Special Triangle
√2
1
π/
4
1
A Special Triangle
sin π/4 = 1/√2
cos π/4 = 1/√2
√2
1
tan π/4 = 1/1 = 1
π/
4
1
Another Special Triangle
2 units
2 units
Another Special Triangle
2
√3
1
Another Special Triangle
π/
6
2
√3
π/
3
1
Another Special Triangle
sin π/6 = 1/2
sin π/3 = √3/2
π/
6
cos π/6 = √3/2
cos π/3 = 1/2
2
√3
tan π/6 = 1/√3
π/
3
1
tan π/3 = √3/1 =√3
445.102
Lecture 4/2
Administration
Last
Lecture
Looking Again at the Unit Circle
Some Other Functions
Equations with Many Solutions
Summary
Lecture 4/2 – Summary
Sine,
cosine and tangent can be seen as
lengths on the Unit Circle that depend on
the angle under consideration.
So sine, cosine and tangent are functions
where the angle is the variable.
For each of these there is a reciprocal
function.
The graphs of these functions can be used
to “see” the solutions of trigonometric
equations
445.102
Before
Lecture 4/2
the next lecture........
Go over Lecture 4/2 in your notes
Do the Post-Lecture exercise p84
Do the Preliminary Exercise p85
See you tomorrow ........