Unpacked the Sine Function

Explore y = R sin (Sx + T) + U

By Pei-Chun Shih

In mathematics, the trigonometry is the study of angles and triangles. Therefore, the trigonometry functions are functions of an angle. There are two ways to define trigonometric functions. The first one is to define them as ratios of the sides of a right triangle and the second one is to define them using a unit circle.

The sine function is one of the basic trigonometry functions. In a unit circle, if θ is an angle measured counterclockwise from the x-axis along the arc, then sinθ is the vertical coordinate of the arc endpoint. This explains why the sine function is periodic with period of 2p. Other properties of the sine function include:

1.                   The graph of the sine function intercepts x-axis at np, where n is an integer.

2.                   The graph of the sine function intercepts y-axis at 0.

3.                   The maximum values of the sine function are y = 1. It occurs when x =  + 2np where n is an integer.

4.                   The minimum values of the sine function are y = -1. It occurs when x =  + 2np where n is an integer.

[y = sin (x)]

Here I am going to examine graphs of the trigonometric sine function: y = R sin (Sx + T) + U for different values of R, S, T, and U.

Given R and S fixed at 1 and U equals 0 and evaluate y = sin (x + T) when T equals 0, ±1, and ±2.

[sin (x), sin (x+1), sin (x-1), sin (x+2), and sin (x-2)]

From the graph above, we can see that the sine function translated horizontally. A horizontal translation of a trigonometric function is called a phase shift. When T > 0, the shift is to the left and when T < 0, the shift is to the right. The value of the phase shift is determined by the value of T when the other two variables R, S are fixed at 1.

More generally, consider the algebraic form of the sine function: y = R sin (Sx + T), where R, S, T­0. In order to solve this function for x, we need to assign a zero to y for which R sin (Sx + T) = 0. Therefore, sin (Sx + T) = 0. It then follows that Sx + T = 0 since sin (0) = 0. Then we get x = -. Therefore, y = 0 when x = -. The phase shift value equals -. From this we can make a conclusion for the phase shift: when T > 0, the graph y = R sin (Sx) shifts to the left for || to form a new equation of y = R sin (Sx + T); when T < 0, the graph y = R sin (Sx) shifts to the right for || to form a new equation of y = R sin (Sx + T).

Given R and S fixed at 1 and T equals 0 and evaluate y = sin (x) + U when U equals 0, ±1, and ±2.

[sin (x), sin (x)+1, sin (x)-1, sin (x)+2, and sin (x)-2]

When a constant is added to a sine function, the graph is shifted vertically. The value of the vertical shift is U. When U > 0, the graph shifts upward and vice versa. Furthermore, the horizontal axis, which the graph oscillates, shifts, too. The new reference line for the graph y = R sin (Sx) + U is the equation y = U.

Let S = 1, T and U equal 0 and examine the equation of the form y = R sin (x) when R > 0.

[sin (x), 1/2sin (x), 2sin (x), and 3sin (x)]

From the graph above, we can observe the vertical expanding and compressing of the parent graph (y = sin x) of the sine functions. Consider an equation of the form y = R sin (x) (R > 0). When R is a number larger than one, the graph vertically expanded compare to the parent graph. When R is a number small than one, the graph vertically compressed. Since the maximum absolute value of sin (x) is 1, the maximum value of R sin (x) is |R| and is called the amplitude of y = R sin (x).

Here we were talking about the situations when R is bigger than zero. What happens when R is smaller than zero? If R is negative, the graph of the sine function upside down as comparing to the graph showed above (see the graph below). However, the amplitude of y = -R sin (x) remains the same as y = R sin (x).

[-sin (x), -1/2sin (x), -2sin (x), and -3sin (x)]

Let us examine our last variable, S, given R equals 1 and T, U equal 0 and examine the equation of the form y = sin (Sx). Let S be any positive number first.

[sin (x), sin (1/2x), sin (2x), sin (3x), and sin (4x)]

Here we see the horizontal expanding and compressing of the parent graph (y = sin x) of the sine functions. Consider an equation of the form y = sin (Sx) (S > 0). When S is a number larger than one, the graph horizontally compressed compare to the parent graph. When R is a number small than one, the graph horizontally expanded. Since the period of the sine function is 2p, then y = sin (Sx) can be written as y = sin (Sx + 2p) according to the definition of periodic function. By the algebraic transformation, we can get

y = sin S(x + ). Therefore, the period of y = sin (Sx) is , where S > 0.

LetÕs discuss about the situations when S is smaller than zero. Just like our observations of negative number R above, the graph of the sine function upside down as comparing to the graph shown above (see the graph below). However, the period of y = sin (-Sx), which equals , remains the same as y = sin (Sx).

[sin (-x), sin (-1/2x), sin (-2x), sin (-3x), and sin (-4x)]

The exploration of the sine function, y = R sin (Sx + T) + U, enables us to have a clearer understanding about its translation, amplitude and period. Next time when seeing a complicate sine function, y = 1.8 sin (x - 8) + 3.3, as shown below

[y = 1.8 sin (1/7x - 8) + 3.3]

You will know that the phase shift of this graph equals -= -(-8)/() = 56 units (where T = -8 and S = ). The vertical shift is 3.3 upward (since U = 3.3 > 0), the amplitude is |1.8|, and the period is= 2p/() = 14p. Conversely, you can write an equation of a sine function with amplitude, period, phase shift, and vertical shift being given. That is to say, you have to find the values of R, S, T, and U and substitute them into the equation y = R sin (Sx + T) + U.