Review We first start with the graph of the basic sine function f (x) = sin (x) The domain of function f is the set of all real numbers. The range of f is the interval [-1,1]. To have a complete picture of why the graph of the sin(x) changes with x as shown above, you may want to go through an interactive tutorial on the trigonometric unit circle. Artificial Girl 3 Han Nari Expansion more.

Graphing f (x) = a*sin(b x + c) We first need to understand how do the parameters a, b and c affect the graph of f (x)=a*sin(bx+c) when compared to the graph of sin(x)? You may want to go through an interactive tutorial on.

The graph of the sine function looks like this: Graph of the sine function. Careful analysis of this graph will show that the graph corresponds to the unit circle. X is essentially the degree measure (in radians), while y is the value of the sine function. The graph of the cosine function looks like this: Graph of the cosine function.

The domain of f is the set of all real numbers. The range of expression bx + c is the set of all real numbers. Therefore the range of sin(bx+c) is [-1,1]. Hence -1 0 -a = a*sin(bx+c) >= a or a 0 0 = x >= 2 p /b. Which is equivalent to 2 p /b 0 and solve for x -c 0, the shift will be to the right. Example 1: f is a function given by f (x) = 2sin(3x - p /2) a - Find the domain of f and range of f. B - Find the period and the phase shift of the graph of f.

C - Sketch the graph of function f over one period. Answer to Example 1 a - The domain of f is the set of all real numbers.

The range is given by the interval [-2, 2]. B - Period = 2 p / b = 2 p /3 Phase shift = - c / b = - (- p /2) / 3 = p /6 c - To sketch the graph of f over one period, we need to find the 5 key points first. Let 3x - p /2 vary from 0 to 2 p in order to have a complete period then find the values of f (x). See table below. 3x- p /2 0 p /2 p 3 p /2 2 p f (x) 0 2 0 -2 0 We now need to find the corresponding values of x.

The first row in the table above gives 0.

• • • In, the sine is a of an. The sine of an acute angle is defined in the context of a: for the specified angle, it is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the ). More generally, the definition of sine (and other trigonometric functions) can be extended to any value in terms of the length of a certain line segment in a. More modern definitions express the sine as an or as the solution of certain, allowing their extension to arbitrary positive and negative values and even to. The sine function is commonly used to model phenomena such as and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughout the year. The function sine can be traced to the functions used in (, ), via translation from Sanskrit to Arabic and then from Arabic to Latin. The word 'sine' comes from a mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.