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How to determine the Sine function from its graph?

Question:

The figure shows a sine wave, y = A sin (Bx + C), in which A > 0, B > 0, |C| < pi/2. Find the analytical expression of the sine wave.

Solution:

Look the sine graph, the maximum value of the sine wave is 3 and the minimum value is -3. So, the amplitude of the sine wave is 3. That is, A = 3.

When x = 2/3, y = 3. When x = 17/12, y = 0. Let T is the minimum positive period. Then T/4 = 17/12 - 2/3

T/4 = 17/12 - 2/3
= 17/12 - (2/3) (4/4)
= (17 - 8)/12
= 9/12
= 3/4
So, T = 3

From definition: T = 2pi/B, then B = 2pi/T = 2pi/3.

So, the function of the sine wave is: y = 3 sin [(2pi/3) x + C].

Now, we determine the initial phase C. Let the point (2/3, 3) be the point P. Because the point P lies on the sine wave, so, the point P satisfy the sine wave function. Now we substitute the point P into the sine function.

3 = 3 sin [(2pi/3) (2/3) + C]
1 = sin (4pi/9 + C)
sin (4pi/9 + C) = 1
4pi/9 + C = pi/2
C = pi/2 - 4pi/9
= (pi/2) (9/9) - (4pi/9) (2/2)
= (9pi - 8pi)/18
= pi/18

So, the initial phase of the sine wave is pi/18.

Therefore, the sine function of the graph is y = 3 sin [(2pi/3) x + pi/18]. Watch the video for more details.