Math 120: Elementary FunctionsCircular Functions

Revisiting the 45-45-90 and 30-60-90 TrianglesExtending the definitions of Trig functions to real numbersGraphing the function y = sin(x)Graphing the function y = cos(x)

Revisiting/Revising Two Famous Triangles

isosceles 45-45-90

equilateral 30-60-90

1

1

1

2

2 23

4

4

6

3

4

4

1 2

1 2

1

3

6

1 2

3 21

Circular Functions - with a change of variable

Add a unit circle - so is equal to the arc length x Since

Thus trig functions are defined for lengths or quantities in the variable x.

x

2 2 1r u v

sin sin x v

cos cos x u

tan tanv

xu

,u v

u

v

Trigonometric Functions of Real Numbers

Let x be any real number and let be the point corresponding to x when the number line wrapped around the unit circle then

A function is periodic if there is a positive number c such that for all x in the domain. The smallest number c is called the period.

2 2 1u v

,P u v

y f x f x c f x

sin cos tan

1 1csc sec cot

vx v x u x uux x xv u v

The 16-Point Unit Circle

1,0

3 1,2 2

31 ,2 2

12, 1

2

0,1

1,0

0, 1

sketch

reference

triangles

Graphing sine(x)

Using the 16 point unit circle sketch the graph of sine(x)

2

1

The 16-Point Unit Circle

1,0

3 1,2 2

31 ,2 2

12, 1

2

0,1

1,0

0, 1

sketch

reference

triangles

Graphing cosine(x)

Using the 16 point unit circle sketch the graph of cosine(x).

2

1