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 Amplitude Amplitude Period Period means how many degrees in one cycle. Amplitude means the distance from the centre to the maximum or minimum, OR (max + min) ÷ 2

f(x) = sin x Period = 360º Amplitude = 1 Period

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; = 120º So B changes the period; the period of the function is (360º ÷ B)

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.

f(x) = A sin Bx + C A = amplitude B = 360º ÷ period C = vertical shift Note: It is exactly the same for sine and cosine. The difference is the where it crosses the y-axis.

What is the equation of this function? Amplitude = 2 so, A = 2 so, B = 3 Period = 120º so, C = -1 Vertical shift = -1 f(x) = 2 sin 3x – 1

What is the equation of this function? Amplitude = 4, going down-up so, A = -4 Period = 720º so, B = 0.5 Vertical shift = 1 so, C = 1 f(x) = -4 sin ½x + 1

What is the equation of this function? Amplitude = 2.5 so, A = 2.5 Period = 240º so, B = 1.5 Vertical shift = 2 so, C = 2 f(x) = 2.5 sin 1.5x + 2