© T Madas

© T Madas

The graphs ofsinx, cosx &

tanx

© T Madas

y

x

0

180

0.26

165

0.5

150

0.71

135

0.87

120

0.97

105

1

90

0.970.870.710.50.260

75604530150

0-0.26-0.5-0.71-0.87-0.97-1-0.97-0.87-0.71-0.5-0.26y

360345330315300285270255240225210195x

y = sin x

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

sinx

© T Madas

y

x

-1

180

-0.97

165

-0.87

150

-0.71

135

-0.5

120

-0.26

105

0

90

0.260.50.710.870.971

75604530150

10.970.870.710.50.260-0.26-0.5-0.71-0.87-0.97y

360345330315300285270255240225210195x

y = cos x

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

cosy x=

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

y

x

0

180

-0.27

165

-0.58

150

-1

135

-1.73

120

-3.73

105

90

3.731.7310.580.270

75604530150

0-0.27-0.58-1-1.73-3.733.731.7310.580.27y

360345330315300285270255240225210195x

y = tan x

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

-5.67-11.4311.435.67-5.67-11.4311.435.67y

280275265260100958580x

y = tan x

asymptotes

© T Madas

8

6

4

2

-2

-4

-6

-8

-720 -540 -360 -180 180 360 540 720

cosy x=

tany x=

siny x=

Plotting the trigonometric functions in one graph

© T Madas

The graphs ofsinx + ccosx + c

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1.5

2

1

-0.5

0.5

y

x

1

180

1.26

165

1.5

150

1.71

135

1.87

120

1.97

105

2

90

1.971.871.711.51.261

75604530150

10.740.50.290.130.0300.030.130.290.50.74y

360345330315300285270255240225210195x

y = sin x + 1

sin 1y x= +

+1

+1

+1+1

+1

+1

+1 +1

+1

+1siny x=

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

2

-1

-2

cos 1y x= -

cosy x=

12

cosy x= +

What is the equation of the purple curve?What is the equation of the green curve?

© T Madas

8

6

4

2

-2

-4

-6

-8

-720 -540 -360 -180 180 360 540 720

Plotting sinx + c for different values of c

sinx

Sinx + 4

Sinx – 8

© T Madas

The graphs ofa sinxa cosx

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

2

-1

-2

y

x

0

180

0.52

165

1

150

1.41

135

1.73

120

1.93

105

2

90

1.931.731.4110.520

75604530150

0-0.52-1-1.41-1.73-1.93-2-1.93-1.73-1.41-1-0.52y

360345330315300285270255240225210195x

y = 2sin x

2siny x=

siny x=y -

str

etc

h

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

2

-1

-2

cosy x=

12cosy x=

What does y = 2cosx look like?What is the equation of the green curve?

2cosy x=

© T Madas

8

6

4

2

-2

-4

-6

-8

-720 -540 -360 -180 180 360 540 720

Plotting a sinx for a = 1, 2, 3, 4, 5 in one graph

sinx2sinx3sinx4sinx5sinx

© T Madas

8

6

4

2

-2

-4

-6

-8

-720 -540 -360 -180 180 360 540 720

Plotting a cosx for a = 1, 2, 3, 4, 5 in one graph

cosx2cosx3cosx4cosx5cosx

© T Madas

The graphs of-a sinx-a cosx-a tanx

© T Madas

y

x

0

360

0.26

345

0.5

330

0.71

315

0.87

300

0.97

285

1

270

0.970.870.710.50.260

255240225210195180

-0.26-0.5-0.71-0.87-0.97-1-0.97-0.87-0.71-0.5-0.260y

1651501351201059075604530150xy = -sin x

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5 -siny x=

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

2

-1

-2

cosy x=

-2cosy x=

What does y = -2cosx look like?

2cosy x=

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

tany x=

-tany x=

© T Madas

8

6

4

2

-2

-4

-6

-8

-720 -540 -360 -180 180 360 540 720

Plotting a sinx for -3 ≤ a ≤ 3 in one graph

Work out each equation

© T Madas

8

6

4

2

-2

-4

-6

-8

-720 -540 -360 -180 180 360 540 720

Plotting a cosx for -3 ≤ a ≤ 3 in one graph

Work out each equation

© T Madas

The graphs ofsin(ax )cos(ax )

© T Madas

y

x

0

180

-0.5

165

-0.87

150

-1

135

-0.87

120

-0.5

105

0

90

0.50.8710.870.50

75604530150

0-0.5-0.87-1-0.87-0.500.50.8710.870.5y

360345330315300285270255240225210195x

y = sin (2x)

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

siny x=

sin(2 )y x=

x - squash

© T Madas

y = sin (2x)

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

siny x=

(3x)

sin(2 )y x=

sin(3 )y x=

© T Madas

Plotting sin(ax) for various values of a1

-1

-360 -270 -180 -90 90 180 270 360

Work out each equation

Plotting cos(ax) for various values of a1

-1

-360 -270 -180 -90 90 180 270 360

© T Madas

The graphs ofsin(x + a) cos(x + a)

© T Madas

y

x

-0.5

180

-0.26

165

0

150

0.26

135

0.5

120

0.71

105

0.87

90

0.9710.970.870.710.5

75604530150

0.50.260-0.26-0.5-0.71-0.87-0.97-1-0.97-0.87-0.71y

360345330315300285270255240225210195x

y = sin(x + 30)

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

siny x=

sin( 30)y x= +

30

30

30

30

3030

30

30

30

30

30

3030

30

30

30

30

30

30

30

30

© T Madas

y

x

-0.87

180

-0.71

165

-0.5

150

-0.26

135

0

120

0.26

105

0.5

90

0.710.870.9710.970.87

75604530150

0.870.710.50.260-0.26-0.5-0.71-0.87-0.97-1-0.97y

360345330315300285270255240225210195x

y = sin(x + 30)

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

siny x= sin( 60)y x= +

60

60

60

60

60

6060

60

60

60

60

60

6060

60

60

60

60

60

60

60

60

© T Madas

y

x

0.87

180

0.97

165

1

150

0.97

135

0.87

120

0.71

105

0.5

90

0.260-0.26-0.5-0.71-0.87

75604530150

-0.87-0.97-1-0.97-0.87-0.71-0.5-0.2600.260.50.71y

360345330315300285270255240225210195x

y = sin(x + 30)

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

siny x= sin( 60)y x= -

60

60

60

60

6060

60

60

6060

60

60

60

60

60

60

60

60

60

60

60

60

60–

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

siny x=sin( 40)y x= +

What are the equations of these curves?

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

siny x= sin( 60)y x= -

What are the equations of these curves?

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

siny x= sin( 70)y x= +

What are the equations of these curves?

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

siny x= sin( 100)y x= -

What are the equations of these curves?

© T Madas

180° 210° 240° 270° 300° 330°0 30° 60° 90° 120° 150° 360°

1

1.5

0.5

-1

-0.5

siny x=

sin( 10)y x= +

sin(x + 90) =cos x

cosy x=

sin x =cos( x – 90)

What are the equations of these curves?

© T Madas

© T Madas

5

4

3

2

1

-1

-2

-3

-4

-5

-600 -500 -400 -300 -200 -100 100 200 300 400 500 600

5

4

3

2

1

-1

-2

-3

-4

-5

-600 -500 -400 -300 -200 -100 100 200 300 400 500 6005

4

3

2

1

-1

-2

-3

-4

-5

-600 -500 -400 -300 -200 -100 100 200 300 400 500 600

5

4

3

2

1

-1

-2

-3

-4

-5

-600 -500 -400 -300 -200 -100 100 200 300 400 500 600

The graphs of four trigonometric functions are shown below.Write the letter of each graph next to the equation which produced it.

A B

C D

CADBLetter

y = sin(2x )y = cosxy = sinxEquation y = sin( x )12

© T Madas

© T Madas

360°0

y

x

1.2

1

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

-1.2

30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330°

The graph of cos2x is shown below for x between 0° and 360°

1. Write down a suitable scale on the x and y axis and hence write down the coordinates of the points A, B, C, D, E and F.

2. Use the graph to find estimates for the values of x for which cos2x = 0.5, for x between 0° and 360°.y = cos2x

The value of sine and cosine lies between…… -1 and 1

A

B

C

D

E F(45,0)

(90,-1)

(135,0)

(180,0)

(225,0) (305,0)

© T Madas

360°0

y

x

1.2

1

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

-1.2

30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330°

The graph of cos2x is shown below for x between 0° and 360°

1. Write down a suitable scale on the x and y axis and hence write down the coordinates of the points A, B, C, D, E and F.

2. Use the graph to find estimates for the values of x for which cos2x = 0.5, for x between 0° and 360°.y = cos2x

A

B

C

D

E F(45,0)

(90,-1)

(135,0)

(180,0)

(225,0) (305,0)

y = ½

cos2x = 0.5

x = 30°x = 150°x = 210°x = 330°

© T Madas

© T Madas

30°0

60° 90° 120° 150° 180° 210° 240° 270° 300° 330° 360°

x

y1.2

1

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

-1.2

y = sinx

The graph of y = sinx is drawn below, for x between 0 and 360°

1. Fill in the scale on the y axis 2. Use the graph to get estimates for the solution of the

equation 4sinx + 1 = 0 for x between 0 and 360°

4 sinx = 0+ 1

4 sinx = -1

sinx = 14

y = - ¼

194° 346°

© T Madas

© T Madas

x

y1.2

1

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

-1.2

90° 180° 270° 360°0

The graph of sinx is shown below for x between 0° and 360°

1. Write down the co ordinates of the points A, B and C.

2. Sketch the graph of sin2x for x between 0° and 360°.

y = sinx

A

B

C

The value of sine and cosine lies between…… -1 and 1

(180,0)

(90,1)

(270,-1)

© T Madas

x

y1.2

1

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

-1.2

90° 180° 270° 360°0

The graph of sinx is shown below for x between 0° and 360°

1. Write down the co ordinates of the points A, B and C.

2. Sketch the graph of sin2x for x between 0° and 360°.

y = sinx

A

B

C

If y = sinx is y = f(x )then y = sin2x is…… y = f(2x )… i.e. squashed by a factor of 2 in x only.(180,0)

(90,1)

(270,-1)

y = sin2x

© T Madas

x

y1.2

1

0.8

0.6

0.4

0.2

-0.2

-0.4

-0.6

-0.8

-1

-1.2

90° 180° 270° 360°0

The graph of sinx is shown below for x between 0° and 360°

1. Write down the co ordinates of the points A, B and C.

2. Sketch the graph of sin2x for x between 0° and 360°.

A

B

C

(180,0)

(45,1)

(135,-1) C(270,-1)

If y = sinx is y = f(x )then y = sin2x is…… y = f(2x )… i.e. squashed by a factor of 2 in x only.

y = sin2x

B(90,0)

© T Madas

© T Madas

The equation of a curve is y = a + sinbx.

The curve passes through A (0,2) and B (5,2½).Find the values of a and b and hence write down the equation of the curve

when x = 0, y = 2

y = a + sinbx

2 = sin(b x 0)a = 2

when x = 5, y = 2½

b = 6

a + sin 0 = 0

2½ = sin(b x 5)2 +½ = sin(5b)

5b = sin-1(½)5b = 30

y = 2 + sin6x

© T Madas

© T Madas

y = a + tanbx

The equation of a curve is y = a + tanbx.

The curve passes through A (0,-1) and B (9,0).Find the values of a and b and hence write down the equation of the curve

when x = 0, y = -1

-1 = tan(b x 0)a = -1

when x = 9, y = 0

b = 5

a + tan 0 = 0

0 = tan(b x 9)-1 +1 = tan(9b)

9b = tan-1(1)9b = 45

y = -1 + tan5x

© T Madas

© T Madas

4

3

2

1

-1

-2

-3

180° 210° 240° 270° 300° 330°30° 60° 90° 120° 150° 360°

The graph of y = p + q sin r x is shown below.

Find the values of p, q and r and hence write down the equation of the curve

when x = 0, y = 1

1 = q sin(r x 0)p = 1

p + sin 0 = 0

1 3 2

© T Madas