Trigonometric Functions

Sine and Cosine Functions

f(x) = sin x and f(x) = cos x

f(x) = sin x and f(x) = cos x

f(x) = sin x & two important ideas

Period

Am

plitudePeriod

Period means how many degrees in one cycle.

Amplitude means the distance from the centre to the maximum or minimum, OR (max + min) ÷ 2

Am

plitude

f(x) = sin x

Period

Period = 360º

Amplitude = 1

Now we will investigate

f(x) = A sin Bx + C

How do A, B and C affect the shape of the graph?

Note: It is exactly the same for sine and cosine, so we will stick just to sine for the start.

f(x) = sin x & f(x) = sin x + 3

f(x) = sin x + 3 & f(x) = sin x – 2

So C moves the curve up and down

f(x) = sin x & f(x) = sin 2x

Period = 180º

f(x) = sin x & f(x) = sin 3x

So B changes the period;

the period of the function is (360º ÷ B)

Period

= 120º

f(x) = sin x & f(x) = 2 sin x

Amplitude = 2

f(x) = sin x & f(x) = 4 sin x

Amplitude = 4

f(x) = sin x & f(x) = -1 sin x

Amplitude = 1

f(x) = sin x & f(x) = -3 sin x

Amplitude = 3

f(x) = sin x & f(x) = A sin x

The A gives the amplitude of the function.

A negative value means the graph goes down – up, not up – down.

A = 4

A = -3

f(x) = A sin Bx + C

A = amplitude

Note: It is exactly the same for sine and cosine.

The difference is the where it crosses the y-axis.

B = 360º ÷ period

C = vertical shift

What is the equation of this function?

Amplitude = 2

Period = 120º

Vertical shift = -1

f(x) = 2 sin 3x – 1

so, A = 2

so, B = 3

so, C = -1

What is the equation of this function?

Amplitude = 4,

going down-up

Period = 720º

Vertical shift = 1

f(x) = -4 sin ½x + 1

so, A = -4

so, B = 0.5

so, C = 1

What is the equation of this function?

Amplitude = 2.5

Period = 240º

Vertical shift = 2

f(x) = 2.5 sin 1.5x + 2

so, A = 2.5

so, B = 1.5

so, C = 2