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 ........