1.1

y = 3 cos x − 2

1.2

y = 1 − sin x

Vertical translation : q = − 2 and the graph

Vertical translation : q = − 1 and the graph

oscillates about y = − 2

oscillates about y = − 1

amplitude = 3 and thus

amplitude = 1 and thus

maximum = − 2 + 3 = 1

maximum = 1 + 1 =

minimum = −2 − 3 = − 5

minimum = 1 − 1 = 0

There is no horizontal translation; p = 0

There is no horizontal translation; p = 0

Period = 360°

Period = 360°

The maxiimum is reached at 0°

The maxiimum is reached at −90°

The minimum is reached at −180° and

The minimum is reached at 90°

at 180°

1.3

y = cos (2x − 60°) = cos 2(x − 30°)

1.4

y = sin (x + 30°)

Vertical translation : q = 0 and the graph

Vertical translation : q = 0 and the graph

oscillates about y = 0; the X-axis

oscillates about y = 0 ; the X-axis

amplitude = 1 and thus maximum = 1

amplitude = 1 and thus maximum = 1

and minimum = − 1

and minimum = − 1

k = 2 and period = 180°

k = 1 and period = 360°

Horizontal translation; p = 30° to the right.

Horizontal translation; p = 30° to the left.

The maxiimum is reached at − 150° and

The maxiimum is reached at 120°

at 30°

The minimum is reached at −60° and

The minimum is reached at − 60°

at 120°

1.5

y = cos (x + 30°) −1

1.6

y = sin (x + 60°) + 1

Vertical translation : q = − 1 and the graph

Vertical translation : q = 1 and the graph

oscillates about y = − 1

oscillates about y = 1

amplitude = 1 and thus

amplitude = 1 and thus

maximum = 1 − 1 = 0

maximum = 1 + 1 = 2

and minimum = −1 − 1 = −2

and minimum = 1 − 1 = 0

k = 1 and period = 360°

k = 1 and period = 360°

Horizontal translation; p = 30° to the left.

Horizontal translation; p = 60° to the left.

The maximum is reached at − 30°

The maximum is reached at 30°

The minimum is reached at 150°

The minimum is reached at − 150°

2.1

y = a sin k(x + p)

2.2

y = a cos k(x + p)

The graph oscillates about y = 0; the X-axis

The graph oscillates about y = 0; the X-axis

and thus q = 0

and thus q = 0

360°

360°

period = 180° and k = ——— = 2

period = 120° and k = ——— = 3

180°

120°

Horizontal translation is 30° to the left; p = +30°.

Horizontal translation is 30° to the right; p = −30°

Equation: y = 1**.** sin 2x + 30°

Equation: y = 2**.** cos 3x − 30°

= sin 2(x + 15°)

= 2 cos 3(x − 10°)

2.3

y = a sin k(x + p)+ q

2.3

y = a cos k(x + p) + q

The graph oscillates about y = 0; the X-axis

The graph oscillates about y = 0; the X-axis

and thus q = 0

and thus q = 0

amplitude = 2 and thus a = 2

amplitude = 3 and the graph

is a cosine graph so that a = − 3

360°

360°

period = 120° and k = ——— = 3

period = 180° and k = ——— = 2

120°

180°

At A(20° ; 0): 3(20° + p) = 0°

At A(15° ; 0): 2(15° + p) = 90°

60° + 3p = 0°

30° + 2p = 90°

p = − 20°

p = 30°

Equation: y = 2**.** sin 3(x − 20°)

Equation: y = − 3**.** cos 2(x + 30°)

= 2 sin 3(x − 20°)

= − 3 cos 2(x + 30°)

2.5

y = a sin k(x + p)+ q

2.6

y = a cos k(x + p) + q

The graph oscillates about y = − 2

The graph oscillates about y = − 2

and thus q = − 2

and thus q = − 2

amplitude = 3 and thus a = 3

amplitude = 3 and thus a = 3

360°

period = 360° and k = 1

period = 180° and k = ——— = 2

180°

There is no horizontal translation and p = 0°

There is no horizontal translation and p = 0°

Equation: y = 3**.** sin (x + 0°) − 2

Equation: y = 3**.** cos 2(x + 0°) − 2

= 3 sin x − 2

= 3 cos 2x − 2

3.1

A(− 150°;0); B(− 135°;0);C(− 45°;0);
D(30°;0); E(45°;0); F(135°;0)

3.2

period = 360°

3.3

sin (x − 30°) = cos 2x

= sin (90° − 2x)

x − 30° = 90° − 2x + n**.**360°

OR

x − 30° = 180° − (90° − 2x) + n**.**360°

3x = 120° + n**.**360°

x − 30° = 180° − 90° + 2x + n**.**360°

x = 40° + n**.**120° ; n ∈ Z

− x = 120° + n**.**360°

x = − 120° − n**.**360°

Solution : x = − 120° −0**.**360°
; 40° − 1**.**120°
; 40° + 0**.**120°
; 40° + 0**.**120°

Solution : x = − 120° ; − 80° ; 40° ; 160°

4.1

amplitude of cosine graph = 2 and thus a = 2; the cosine graph reaches 0 at 60° in stead of 90°

so that the graph is shifted 30° to the left and thus b = 30°

the sine graph oscillates about y = 1 so that c = 1 ; the period = 180° and thus d = 2.

4.2

f(x) = 2 cos (x + 30°)

f(0) = 2 cos (0 + 30°) = 2 cos 30° = √3

4.3.1

x = 45° and x = −135°

4.3.2

f and g intersect at(−90° ; 1) and at (13,5° ; 1,45);
hence solution: −90° ≤ x ≤ 13,5°

4.4

y = 2 cos x

5.1

amplitude of cosine graph = 1 and thus a = 1; the cosine graph reaches 1 at 30° in stead of 0°

so that the graph is shifted 30° to the right and thus b = −30°

the sine graph has a period = 360° and thus c = 1.

5.2.1

Range : − 0,5 ≤ y ≤ 1

5.2.2

30° < x ≤ 90°

5.2.3

−90° ≤ x ≤ −60° and x = 0°

5.3

cos (x − 30°) = sin x

= cos (90° − x)

∴ x − 30° = 90° − x + n360°
n ∈ Z

∴ 2x = 120° + n360°

∴ x = 60° + n360°

∴ x = 60°

If g(x) > f(x) then 60° < x ≤ 90°

6.1

amplitude = 2 and hence c = 2 and period = 360° and thus d = 1

6.2

amplitude = 1 and thus a = 1; f(60°) = 1 and thus graph is shifted 60° to the right so that b = − 60°

6.3

range: − 2 ≤ y ≤ 2

6.4

− 120° ≤ x ≤ − 30°

6.5

y = 2 sin (x − 30°)