445.102 mathematics 2 module 4 cyclic functions lecture 2 reciprocal relationships
TRANSCRIPT
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
AdministrationLast Lecture Looking Again at the Unit Circle Some Other Functions Equations with Many Solutions Summary
RadiansA mathematical measure of angle is defined using the radius of a circle.
1 radian
sin(ø)
øsin(ø)
1
Post-Lecture Exercise1 45° = π/4 radians 60° = π/3 radians
80° = 4π/9 radians 2 full turns = 4π radians
270° = 3π/2 radians
2 π radians = 180° 3 radians = 171.9°
6π radians = 3 turns
3 f(x) = sin x is an ODD function.
4 f(2.5) = 0.598 f(π/4) = 0.707
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
5 The domain of f(x) = sin x is the Real Numbers
6 The domain of the inverse function is –1 ≤ x ≤ 1
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)
-0.50
-1.00
0.50
1.00
3π 4π
f(ø) = sin ø
π 2π-π-2π
Preliminary Exercise
-0.50
-1.00
0.50
1.00
3π 4π
f(ø) = sin ø
π 2π-π-2π
445.102 Lecture 4/2
Administration Last LectureLooking Again at the Unit Circle Some Other Functions Equations with Many Solutions Summary
C(ø)
øC(ø)
1
cos(ø)
øcos(ø)
1
tan(ø)
ø
tan(ø)
1
Constructions on the Unit Circle
øcos(ø)
1sin(ø)
tan(ø)
The Cosine Function(Many Rotations)
-0.50
-1.00
0.50
1.00
π 2π-π-2π 3π 4π
f(ø) = cos ø
The Tangent Function(Many Rotations)
-0.50
-1.00
0.50
1.00
π 2π-π-2π 3π 4π
f(ø) = tan ø
445.102 Lecture 4/2
Administration Last Lecture Looking Again at the Unit CircleSome Other Functions Equations with Many Solutions Summary
The Secant Function
secant
sec ø/1 = sec ø = 1/cos ø
cos(ø)
1
sec 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. What angle has a sine of 0.25 ? 2. What angle has a tangent of 3.5 ? 3. What angle has a cosine of –0.4 ? 4. What is sec π/2 ?
5. What is cot 5π/3 ?
6. What is arctan 10 ?
445.102 Lecture 4/2
Administration Last Lecture Looking Again at the Unit Circle Some Other FunctionsEquations with Many Solutions Summary
An Equation
-0.50
-1.00
0.50
1.00
π 2π-π-2π 3π 4π
f(ø) = cos ø
2cos ø – 0.6 = 02cos ø = 0.6cos ø = 0.3
An Example ....
4sin ø + 3 = 14sin ø = –2sin 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 = 14sin ø = –2sin 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
1
1
√2
π/4
A Special Triangle
1
1
√2
π/4
sin π/4 = 1/√2
cos π/4 = 1/√2
tan π/4 = 1/1 = 1
Another Special Triangle
2 units
2 units
Another Special Triangle
2√3
1
Another Special Triangle
2
π/3
π/6
√3
1
Another Special Triangle
2
π/3
π/6
√3
1
sin π/6 = 1/2
cos π/6 = √3/2
tan π/6 = 1/√3
sin π/3 = √3/2
cos π/3 = 1/2
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 SolutionsSummary
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 Lecture 4/2
Before 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 ........