This is called the horizontal displacement of also known as the phase shift.

Transcription

1 sin (x) GRAPHS OF TRIGONOMETRIC FUNCTIONS Definitions A function f is said to be periodic if there is a positive number p such that f(x + p) = f(x) for all values of x. The smallest positive number p for which this equality holds true is called the period of f. A function is said to be periodic if its values repeat over a certain interval. The graph of a periodic function over a single period is called a full wave or a cycle of the graph. The amplitude of a graph of a function f is defined to be the absolute value of onehalf the difference between its greatest and lowest function values. The amplitude indicates the height of the graph. The displacement of a graph of a function f is defined as the amount by which the graph is shifted from its basic curve either to the right or to the left. This is called the horizontal displacement of also known as the phase shift. The translation of a graph of a function f is defined as the amount by which the graph is shifted up or down from its basic curve. This is called the vertical translation. The Sine Function If we assign the following values for x we would obtain its corresponding sine values as follows: x Plotting these points above, we can sketch the graph of the sine function

2 y sin( x ) Properties of the sine function 1. The sine function is periodic, with p. (This tells us that the graph repeats itself every -interval.. From x 0 to x, the curve rises from y = 0 to y = 1. From x to x, the curve falls from y = 1 to y = 0. From x to x, the curve continues to fall from y = 0 to y = -1. From x to x, the curve rises from y = -1 to y = 0.. The amplitude of the sine function is 1, since 1 sin( x ) 1.. The zeros of the sine function y sin( x ) are 0,,, etc or in general k where k is any integer. 5. The sine function is an odd function, that is, sin (-x) = -sin(x) for any value of x.. Based from the graph of the sine function and the nature of its definition, the domain of the sine function is the set of all real numbers and its range is the interval [-1, 1].

3 Cos (x) The Cosine Function If we assign the following values for x we would obtain its corresponding cosine values as follows: x Plotting these points above, we can sketch the graph of the cosine function y cos( x ) Properties of the cosine function 1. The cosine function is periodic, with p. (This tells us that the graph repeats itself every -interval.. From x 0 to x, the curve falls from y = 1 to y = 0. From x to x, the curve continues to fall from y = 0 to y = -1. From x to x, the curve rises from y = -1 to y = 0. From x to x, the curve continues to rise from y = 0 to y = 1.. The amplitude of the cosine function is 1, since 1 sin( x ) 1.

4 . The zeros of the sine function y cos( x ) are the odd multiples of or in general n where n = 1,, 5, The cosine function is an even function, cos (-x) = cos (x) for any value of x.. Based from the graph of the cosine function and the nature of its definition, the domain of the cosine function is the set of all real numbers and its range is the interval [-1, 1]. The Tangent Function y tan( x ) sin( x ) If we recall that we defined the tangent function as y = tan (x) = is defined only cos( x ) for all values of x where cos (x) 0, that is, for x k, where k is an integer. We can observe that when x increases from to 0, the value of the tangent function

5 approaches to 0 and as x increases from 0 to the value of the tangent function increases without bound or becomes infinitely large. Moreover, as the value of x approaches (through values of x greater than ), tan (x) becomes infinitely small or decreases without bound. The vertical lines x and x which are being approached by the curves, but are never touched are called the vertical asymptotes of the curve. Properties of the tangent function 1. The tangent function has period p.. The domain of the tangent function is the set of all real numbers x except x k, where k is an integer.. In general, the vertical asymptotes of the tangent function are the lines that are of the form x k, where k is an integer. The Cotangent Function y cot( x )

6 cos( x ) Recall the definition of the cotangent function, cot( x ), and we restrict that sin( x ) sin(x) must not be equal to zero, that is x k where k is an integer. Properties of the cotangent function 1. Since the tangent and the cotangent functions are reciprocal functions, the period of the cotangent function is also.. We also note that the asymptotes of the cotangent function are the vertical lines x = k where k is an integer.. The domain of the cotangent function is the set of all real numbers except the values of x = k where k is an integer.. The range of the cotangent function is the set of all real numbers, (-, + ). The Secant and Cosecant Function y sec( x )

7 1 The graph of the secant function y sec( x ) is related to the graph of the cosine cos( x ) function y cos( x ). And that the zeros of the cos(x) are the asymptotes of y = sec(x), k that is, the vertical asymptotes of the function y = sec(x) are the lines x where k is an odd integer. y sec( x ) 1 If we recall the definition of the cosecant function we have, y csc( x ), thus we sin( x ) restrict sin(x) 0, that is, x k, where k is an integer. Since the sine and cosecant are reciprocal functions, the period of the cosecant function is also.