11B Topic 4_2. Model Find the exact value of: (a) (b) (c) We are now familiar with the Unit Circle,...

11B Topic 4_2

ModelFind the exact value of: (a)

(b)

(c)

300sin

225tan

225cos

We are now familiar with the Unit Circle, but to answer these questions we will

need to use the Unit Triangles as well…

45

1

1

2

1sin 45

21

cos 452

tan 45 1

60

30

1

2 3

3sin 60

21

cos602

tan 60 3

1sin 30

2

3cos30

21

tan 303


(b)

(c)

300sin

225tan

225cos

225cos45cos

21

45

45

1

1

2

60

30

1

2 3


(b)

(c)

300sin

225tan

225cos

225tan45tan

1

45

45

1

1

2

60

30

1

2 3


(b)

(c)

300sin

225tan

225cos

300sin60sin

23

60

45

1

1

2

60

30

1

2 3

Now let’s do the same again, using radians

Scootle: 11 Maths B folder• Topic 4 (PWJXSR)• Trig Radians

http://www.scootle.edu.au/ec/p/home


(b)

(c)

300sin

225tan

225cos4

5

3

5

45

1

1

2

60

30

1

2 3

4

6

3


(b)

(c)

3

5sin

4

5tan

4

5cos

4

5cos

4cos

21

4

1

1

2

1

2 3

4

6

3


(b)

(c)

4

5tan

4tan

1

3

5sin

4

5tan

4

5cos

4

1

1

2

1

2 3

4

6

3


(b)

(c)

3

5sin

3sin

23

3

5sin

4

5tan

4

5cos

3

1

1

2

1

2 3

4

6

3

Exercise

NewQ P 307 Set 9.2

Numbers 1, 2, 8-11

For Homework, look at…Scootle: 11 Maths B folder• Topic 4 (PWJXSR)• Trig degrees• Trig radians


For Homework, look at…Scootle: 11 Maths B folder• Topic 4 (PWJXSR)• Trigonometry: assessment


You should now be familiar with the general shape of the three major trignometric graphs

The general shapes of the three major trigonometric graphs

y = sin x

y = cos x

y = tan x

5. Significance of the constants A,B and D on the graphs of…

y = A sin[B(x + C)] + D

y = A cos[B(x + C) ]+ D

2. Open the file y = sin(x)(Excel File)

Scootle: 11 Maths B folder• Topic 4 (PWJXSR)• Eagle Cat

1. Open the file y = Asin[B(x+C)]+d(Autograph file)


y = A cos B(x + C) + D

A: adjusts the amplitude

B: determines the period (T). This is the distance taken to complete one cycle where T = 2/B. It therefore, also determines the number of cycles between 0 and 2.

C: moves the curve left and right by a distance of –C (only when B is outside the brackets)

D: shifts the curve up and down the y-axis

Graph the following curves for 0 ≤ x ≤ 2a) y = 3sin(2x)b) y = 2cos(½x) + 1c) y = sin[2(x + )]d) y = 4cos[2(x - /2)] – 3

Exercise

NewQ P 318 Set 9.4 1 - 6

6. Applications of periodic functions

Challenge Question (1)

High tide is 4.5 m at midnightLow tide is 0.5m at 6am

i) Find the height of the tide at 7pm?ii) Between what times will the tide be greater

than or equal to 3m?

Use y = A cos B(x+C) + D

i) Find “A”

Tide range = 4.5 - 0.5 = 4

A = 2

y = 2cos B(x+C) + D

iii) Find “B”

Period = 12

ii) Find “D”

D = 4.5 – 2 = 2.5

y = 2cos B(x+C) + 2.5

2Period=

2

12

6

2cos 2.56

B

B

B

y x C

iv) Find “C”

We can see from the graph that no C-value is needed

High tide is 4.5 m at midnight Low tide is 0.5m at 6ami) Find the height of the tide at 7pm?ii) Between what times will the tide be greater than or equal

to 3m?

2cos 2.56

xy

2cos 2.56

xy

By use of TI calculator…

i) What is the tide height at 7pm?

• Graph using suitable windows• 2nd Calc option 1. Value• Enter 19• Answer = 0.77m (2D.P.)

ii) Tide above 3m• Add y = 3 to the graph• 2nd Calc option 5. Intersect• Follow prompts• Answer = • MN – 2:31am • 9:29am – 2:31pm• 9:29pm – MN

Challenge Question (2)

High tide of 4.2m occurs in a harbor at 4am Tuesday and the following low tide of 0.8m occurs 6¼ hours later. If a ship entering the harbor needs a minimum depth of water of 3m, what times on Tuesday can this vessel enter?

Model: The graph below shows the horizontal displacement of a pendulum from its rest position over time:

(a) Find the period and amplitude of the movement.(b) Predict the displacement at 10 seconds.(c) Find all the times up to 20 sec when the displacement will be 5 cm to the

right (shown as positive on the graph)

X

Y

1 2 3 4 5

-8

-6

-4

-2

2

4

6

8

0




X

Y

1 2 3 4 5

-8

-6

-4

-2

2

4

6

8

0

Period = 4.5 - 0.5

= 4 sec




X

Y

1 2 3 4 5

-8

-6

-4

-2

2

4

6

8

0

Amplitude = 8




X

Y

1 2 3 4 5

-8

-6

-4

-2

2

4

6

8

0

Since the period = 4 sec

Displacement after 10 sec will be the same as displacement after 2 sec

= 5.7cm to the left




X

Y

1 2 3 4 5

-8

-6

-4

-2

2

4

6

8

0

Displacement= 5cm

t = 1.1

3.9 7.9, 11.9, 15.9, 19.9

5.1, 9.1, 13.1, 17.1

Exercise

NewQ P 179 Set 5.2 1,3

Model: Find the equation of the curve below.

X

Y

1 2 3 4 5 6 7 8 9 10

-2

2

0

Amplitude = 2.5 y = a sin b(x+c)


X

Y

1 2 3 4 5 6 7 8 9 10

-2

2

0

Amplitude = 2.5 y = 2.5 sin b(x+c)

Period = 6

Period = 2/b 6 = 2/b

b = /3


X

Y

1 2 3 4 5 6 7 8 9 10

-2

2

0

Amplitude = 2.5 y = 2.5 sin /3(x+c)

Period = 6


b = /3

Phase shift = 4 ()

so c = -4


X

Y

1 2 3 4 5 6 7 8 9 10

-2

2

0

Amplitude = 2.5 y = 2.5 sin /3(x-4)

Period = 6


b = /3

Phase shift = 4 ()

so c = -4

Exercise

NewQ P 183 Set 5.3 1,4

Exercise 5.3 pg 183, No.4

Find the equation of the curve below in terms of the sin function and the cosine

function.

11B Topic 4_2. Model Find the exact value of: (a) (b) (c) We are now familiar with the Unit Circle,...

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