Circular Functions Revision

Errors made by students with Circular Functions

1. Not showing working2. Not knowing the rules and Exact Values3. Poor understanding and reasoning ie.Splitting the angle and its coefficient

when they should be kept together eg. sin 2θ = 1 then sin θ = 1/2 4. Not checking solutions are in the required

domain5. Errors in calculator use – Use of Radian

and Degree Mode

NO !!!sin 2θ sin θ

When do you use degree mode and when radian mode ?

o Circular Functions and Trigonometry – If you are taking the sine or cosine of an

angle, then as a general rule: If it is in radians – use RADIAN MODE If it is in degrees – use DEGREE MODE On examination papers – radian

measure should be assumed unless otherwise indicated.

For example x → sin x˚

Can you convert angles?From degrees to radians?

From radians to degrees?

Degrees Radians

180

180

Have you memorised the exact values?

Angle 0 or c

sin cos tan

0, 2 , 3600 0 1 0

6

300

1

2 3

2

1 3

33

4

450

1 2

22

1 2

22

1

3

600

3

2

1

2

3

2

900

1

0

undefined

1800

0

– 1

0

3

2

2700

– 1

0

undefined

Do you know how to use the CAST circle?

(Quadrant 2) S A (Quadrant 1) (Quadrant 3) T C (Quadrant 4)

P()P(-)

P(+) P(2-)

0 or 2

2

2

/2

3/2

Are you familiar with the basic shapes of: y = sinx a = 1Min is -1 and max is 1.

Use this information to draw horizontal lines.

Period = 2

1 – 1

y

x

2

2

32

32

2

2

1

1

– 1

– 1 It starts at the mean line

Then at max

Then at the mean line

Then at min

Then at the mean line

What about this? y = cosx a = 1Min is -1 and max is 1.

Use this information to draw horizontal lines.

Period = 2

1 – 1

y

x

2

2

32

32

2

2

1

1

– 1

– 1 It starts at max

Then at max

Then at the mean line Then at min

Then at the mean line

And this one…. y = tanx

y

x

2

2

32

32

2

2

Can you find the asymptotes of tan(bx)? Let Thus

Work out the period of tan(bx)

Which is

Others can be found by adding integer multiples of the period

2bx

2x

b

b

You must know how to solve trigonometric equations in a given domain Draw a CAST circle Tick the two quadrants in which the given

function is positive or negative. Find the first quadrant angle, irrespective of

the sign. Find the first two solutions between x = 0 and

x = 2 (use the appropriate sine, cosine or tangent symmetry property).

If more solutions are required: Repeatedly add (or subtract) the period to the

two solutions as many times as required, noting solutions after each addition or subtraction.

Stop when all solutions within the specified domain are found.

You must know the important features of the sine and cosine graphsy = a sin(bx) + c and y = a cos(bx) + c‘a’ is the dilation factor from the x-axis.

The absolute value of ‘a’ gives the amplitude of the graph.

‘1/b’ is the dilation factor from the y-axis. The period of the graph is ‘c’ is the translation factor which moves

the graph up or down.

360 2or

b b

The maximum value of the function occurs when sin(bx) and cos(bx) = +1.

The minimum value of the function occurs when sin(bx) and cos(bx) = -1.

Range = [-a + c, a + c]

Problem 1: Heart RateThe heart rate of an athlete during a

particular hour of a workout was carefully monitored.

t15 30 45 60

H

40

80

120

160

Reading from the graphWhat is the initial

heart rate?110 beats/min.What is the

minimum heart rate?60 beats/min.

t15 30 45 60

H

40

80

120

160

Finding the rule for this Heart Rate graph

Sine function: H = a sin(bt) + cDetermine the values of a, b and c.To determine the amplitude we subtract the

minimum point. from the maximum point and divide by 2 .

a=(160-60)/2 = 50

t15 30 45 60

H

40

80

120

160

max min

2a

This is the amplitude

The mean line is at H = 110The graph has been translated up 110c = 110Period = 60The period helps us find the ‘b’ value.

t15 30 45 60

H

40

80

120

160

This is the mean

line

The graph completes its cycle

in 60 seconds. Thus the period is 60.

The period is 60 =

b = 6 or

H = or

When modelling with trig functions we generally work with radians unless otherwise specified.

50sin 11030

tH

360 2or

b b

360 2or

b b

30

50sin 6 110t

Problem 2: Bungee JumpingThe height of a bungee jumper, h metres,

above a pool of water at any time t seconds after jumping is described by the function: h(t) = 20 cos(0.8t) + 20

What is the initial height of the bungee jumper?

When, if at all, does the bungee jumper first touch the water?

Assuming the cord is elastic: how long will it be before she returns to the lowest point?

Initial Height.The initial height will occur when t = 0

secsSubstituting t = 0 into the given equation h(t)

= 20 cos(0.8t) + 20 will give us the initial height

h(0) = 20 cos(0) + 20 h(0) = 20 x 1 + 20 = 40 metres above the

pool of water.

Will she hit the pool of water?

The minimum of the graph will occur when the cos value is -1.

The height of the bungee jumper would then be: 20 x (-1) + 20 = 0.

She will hit the water!

When will she hit the water?At the minimum pointWhen cos(0.8t) = -1cos(0.8t) = -1 when 0.8t = t = 3.927The bungee jumper will first touch the water after 4 seconds (to the nearest second).

How long will it be before she returns to the lowest point?y

x5 10 15 20

10

20

30

40

50

– 10

From this sketch we can see that she will hit the water again somewhere between 11 and 12 seconds.

Solving Trigonometric Equationscos(0.8t) = -1.0.8t = and 3Therefore t = 11.79 secondsThis will be 8 seconds (to the nearest second) after the first time.

AlternativelyTo find the time that the next minimum

occurs we could have added on the period of the graph to 4 seconds.

The period is found by Thus the period is 8 seconds.The next time the bungee jumper will touch

the water will be 8 seconds later.

2 2

0.8b

Applications of sine and cosine graphsBasic graph types are

y = a sin(bx) + c and y = a cos(bx) + cTo find the maximum value of a function,

replace sin(bx) or cos(bx) with +1To find the minimum value of a function,

replace sin(bx) or cos(bx) with -1Initial values occur at t = 0A sketch graph may provide greater

understanding.