Trigonometric Graphs. Inverness College. September 29, 2010

Transcription

1 September 29, 2010

2 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 these functions. You will need to be able to recognise these functions and sketch their graphs.

3 sin(x)

4 cos(x)

5 tan(x)

6 How do we plot them?

7 How do we plot them? We use the critical points zeros, maximums and minimums.

8 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.

9 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(360 o ) but the period of tan(x) is only π(180 o ).

10 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(360 o ) but the period of tan(x) is only π(180 o ). Next we need to know where everything occurs. Function Zeros Maximum Minimum sin(x) 0, π(180 o ), 2π(360 o π ) 2 (90o 3π ) 2 (270o ) π cos(x) 2 (90o ), 3π 2 (270o ) 0, 2π(360 o ) π(180 o ) tan(x) 0, π(180 o ), 2π(360 o ) undefined undefined

11 More complex functions We will now look at more complex trigonometric functions.

12 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.

13 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).

14 y = A sin(x) and y = A cos(x) A is the Amplitude.

15 y = A sin(x) and y = A cos(x) A is the Amplitude. Maximum increases to A and minimum decreases to A.

16 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).

17 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).

18 y = A sin(x) and y = A cos(x) For sketching plot critical points (zeros, max. and min.). Make sure you have the correct period. Remember the axes. Try the questions on pages 10 & 11 of your notes.

19 y = sin(nx), y = cos(nx) and y = tan(nx) sin(nx), cos(nx) n periods in 2π(360 o ). tan(nx) n periods in π(180 o ).

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

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

22 y = sin(nx), y = cos(nx) and y = tan(nx) sin(nx), cos(nx) n periods in 2π(360 o ). tan(nx) n periods in π(180 o ).

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

24 y = sin(nx), y = cos(nx) and y = tan(nx) Sketching Work out critical points. Divide x coordinates by n.

25 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.

26 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.

27 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.

28 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) 45 o to the right.

29 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.

30 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) 30 o to the left.

31 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.

32 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) 60 o to the left.

33 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.

34 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.

35 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.

36 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.

37 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.

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