Condensed Presentation Handouts on Trigonometric Graphs

Trigonometric Graphs

Inverness College

September 29, 2010

Inverness College Trigonometric Graphs

Simple Trigonometric Functions

� We begin with the standard trigonometric functions sin(x),cos(x) and tan(x).

� We will then move onto more complex versions of thesefunctions.

� You will need to be able to recognise these functions andsketch their graphs.


sin(x)


cos(x)


tan(x)


How do we plot them?

� We use the critical points – zeros, maximums and minimums.

� sin(x) and cos(x) have maximums of 1 and minimums of -1.

� The period of sin(x) and cos(x) is 2π(360o) but the periodof tan(x) is only π(180o).

� Next we need to know where everything occurs.

Function Zeros Maximum Minimumsin(x) 0, π(180o), 2π(360o) π

2 (90o) 3π

2 (270o)cos(x) π

2 (90o), 3π

2 (270o) 0, 2π(360o) π(180o)tan(x) 0, π(180o), 2π(360o) undefined undefined


More complex functions

We will now look at more complex trigonometric functions.

� y = A sin(x) and y = A cos(x). A represents the amplitude.

� y = sin(nx), y = cos(nx) and y = tan(nx).

� y = A sin(x± b), y = A cos(x± b) and y = A tan(x± b).

� y = A sin(x)± b and y = A cos(x)± b.

We begin with y = A sin(x) and y = A cos(x).


More complex functions

We will now look at more complex trigonometric functions.

� y = A sin(x) and y = A cos(x). A represents the amplitude.

� y = sin(nx), y = cos(nx) and y = tan(nx).

� y = A sin(x± b), y = A cos(x± b) and y = A tan(x± b).

� y = A sin(x)± b and y = A cos(x)± b.

We begin with y = A sin(x) and y = A cos(x).


y = A sin(x) and y = A cos(x)

� A is the Amplitude.� Maximum increases to A and minimum decreases to −A.� Plot of 3 sin(x) (blue dashed line) and sin(x).



� A is the Amplitude.� Maximum increases to A and minimum decreases to −A.� Plot of 3 cos(x) (blue dashed line) and cos(x).



For sketching – plot critical points (zeros, max. and min.). Makesure you have the correct period. Remember the axes.

Try the questions on pages 10 & 11 of your notes.


y = sin(nx), y = cos(nx) and y = tan(nx)

� sin(nx), cos(nx) – n periods in 2π(360o).� tan(nx) – n periods in π(180o).� Plot of sin(2x) (blue dashed line) and sin(x).



� sin(nx), cos(nx) – n periods in 2π(360o).� tan(nx) – n periods in π(180o).� Plot of cos(2x) (blue dashed line) and cos(x).



� sin(nx), cos(nx) – n periods in 2π(360o).� tan(nx) – n periods in π(180o).



Sketching� Work out critical points. Divide x coordinates by n.� Periods: cos(x), sin(x) – 2π(360o)

n , tan(x) – π(180o)n .

� Plot them. Don’t forget your axes. Join them up.


y = A sin(x± a), y = A cos(x± a), and y = A tan(x± a)

� Move graphs left +a or right −a along the x–axis.� Easier in degrees.� sin(x− 45) (blue dashed line) moves sin(x) 45o to the right.



� Move graphs left +a or right −a along the x–axis.� Easier in degrees.� 6 cos(x+30) (blue dashed line) moves 6 cos(x) 30o to the left.



� Move graphs left +a or right −a along the x–axis.� Easier in degrees.

� tan(x+ 60) moves tan(x) 60o to the left.



� Move graphs left +a or right −a along the x–axis.� Easier in degrees.� tan(x+ 60) moves tan(x) 60o to the left.



Sketching� Work out critical points. Move them left (x+ a) or right

(x− a) by a.� Plot them. Don’t forget your axes. Join them up.


y = A sin(x)± b and y = A cos(x)± b

� Move the whole graph up +b or down −b the y–axis by b.� 3 sin(x) + 2 (blue dashed line) moves sin(x) up the y–axis by

2.


y = A sin(x)± b and y = A cos(x)± b

� Move the whole graph up +b or down −b the y–axis by b.� 4 cos(x)− 3 (blue dashed line) moves cos(x) up the y–axis by

3.


Trigonometric Graphs

� Work through Exercise 4A� Exercise 4B: 1a), c), 2e), 3 and 4.� Exercise 4C.


Condensed Presentation Handouts on Trigonometric Graphs

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