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.

Inverness College Trigonometric Graphs

sin(x)

Inverness College Trigonometric Graphs

cos(x)

Inverness College Trigonometric Graphs

tan(x)

Inverness College Trigonometric Graphs

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

Inverness College Trigonometric Graphs

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).

Inverness College Trigonometric Graphs

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).

Inverness College Trigonometric Graphs

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).

Inverness College Trigonometric Graphs

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 cos(x) (blue dashed line) and cos(x).

Inverness College Trigonometric Graphs

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

Inverness College Trigonometric Graphs

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).

Inverness College Trigonometric Graphs

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 cos(2x) (blue dashed line) and cos(x).

Inverness College Trigonometric Graphs

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).

Inverness College Trigonometric Graphs

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

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.

Inverness College Trigonometric Graphs

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.

Inverness College Trigonometric Graphs

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.� 6 cos(x+30) (blue dashed line) moves 6 cos(x) 30o to the left.

Inverness College Trigonometric Graphs

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.

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

Inverness College Trigonometric Graphs

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

Inverness College Trigonometric Graphs

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

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.

Inverness College Trigonometric Graphs

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.

Inverness College Trigonometric Graphs

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.

Inverness College Trigonometric Graphs

Trigonometric Graphs

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

Inverness College Trigonometric Graphs