4.2 Trigonometric 4.2 Trigonometric Function: The Unit circleFunction: The Unit circle

The Unit Circle

A circle with radius of 1

Equation x2 + y2 = 1

sin,cos

0,1

1,0

0,1

1,0

The Unit Circle with Radian Measures

2

Do you remember 30º, 60º, 90º triangles?

Now they are really! Important

Do you remember 30º, 60º, 90º triangles?

Now they are really! Important

Even more important

Let 2a = 1

Do you remember 30º, 60º, 90º triangles?

Let 2a = 1

2

130

2

330

Sin

Cos

Do you remember 30º, 60º, 90º triangles?

1

2

1

3

2

360

2

160

Sin

Cos

Do you remember 45º, 45º, 90º triangles?

When the hypotenuse is 1

The legs are 2

2

2

245

2

245

Sin

Cos 1

2

2

2

2

Some common radian measurements

These are the Degree expressed in Radians

360

445

630

23602

3270

1802

90

The Unit Circle: Radian Measures and Coordinates

2

The Six Trig functions

adjacent

opposite

b

aTan

hypotenuse

opposite

c

aSin

hypotenuse

adjacent

c

bCos

CotTan

CscSin

SecCos

tan

1sin

1cos

1

Cos

SinTan

Why does the book use “t” for an angle?

Since Radian measurement are lengths of an arc of the unit circle, it is written as if the angle was on a number line.

Where the distance is “t’ from zero.

Later when we graph Trig functions it just works better.

Lets find the Trig functions if

Think where this angle is on the unit circle.

3

2

3

21

23

3

2

2

3

3

2

2

1

3

2

Tan

Sin

Cos

3

2

Cos

SinTan

2

3,

2

1

Find the Trig functions of

Think where this angle is on the unit circle.

3

21

23

3

2

2

3

3

2

2

1

3

2

Tan

Sin

Cos

3

2

3

3

3

1

3

2

3

32

3

2

3

2

21

2

3

2

Cot

Csc

Sec

How about

4

4

2

2,

2

2

2

2,

2

2

1

22

22

4

2

2

4

2

2

4

Tan

Sin

Cos

There are times when Tan or Cot does not exist.

At what angles would this happen?

There are times when Tan or Cot does not exist.

2

3,

2

If think of the domain of the trig functions, there are some limits.

Look at the unit circle. If x goes with Cos, then what are the possible of Cos?

It is the same with

Sin?

Definition of a Periodic Function

A function “f” is periodic if there exist a positive real number “ c” such that

f(t + c) = f(t) for all values of “t”.

The smallest “c” is called the period.

Even Function ( Trig. )

Cos (- t) = Cos (t) and Sec( -t) = Sec (t)

Also

Sin(-t) = -sin (t) and Csc (-t) = - Csc (t)

Tan(-t) = -Tan (t) and Cot(-t) = - Cot (t)

HomeworkHomework

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##1, 5, 9, 13, 17, 1, 5, 9, 13, 17,

21, 25, 29, 33, 21, 25, 29, 33,

37, 41, 45, 48, 37, 41, 45, 48,

52, 59, 6852, 59, 68

HomeworkHomework

Page 278- 279 Page 278- 279

## 2, 8, 12, 16, 2, 8, 12, 16,

20, 24, 28, 32, 20, 24, 28, 32,

36, 40, 44, 49, 36, 40, 44, 49,

58, 6158, 61