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Lecture

# unit 5 to 7.docx

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School
University of Toronto Scarborough
Department
Mathematics
Course
MATA30H3
Professor
Ken Butler
Semester
Winter

Description
Unit 5: Trigonometric Functions  Graphs of Sine, Cosine, and Tangent Functions  The graphs of y = sin x, y = cos x, and y = tan x are periodic.  The graphs of y = sin x and y = cos x are similar in shape and have an amplitude of 1 and a period of 2π  The graph of y = sin x can be transformed into graphs modeled by equations of the form y = sin x + c, y = sin (x – d), and y = sin kx. Similarly, the graph of y = cos x can be transformed into graphs modeled by equations of the form y = cos x + c, y = acos x, y = cos (x – d), and y = cos kx.  The graph of y = tan x has no amplitude because it has no maximum or minimum values. It is undefined at values of x that are odd multiples of π/2, such as π/2 and 3π/2.  The graph becomes asymptotic as the angle approaches these values from left and the right. The period of the function is π.  Graphs of Reciprocal Sine, Cosine, and Tangent Functions  The graphs of y = csc x, y = sec x, and y = cot x are periodic. They are related to the primary trigonometric functions as reciprocal graphs.  Reciprocal trigonometric functions are different from inverse trigonometric functions.  csc x means 1 / sin x, while sin x asks you to find an angle that has a sine ratio equal to x. -1  sec x means 1 / cos x, while cos x asks you to find an angle that has a cosine ratio equal to x.  cot x means 1 / tan x, while tan x asks you to find an angle that has a tangent ratio equal to x.  Sinusoidal functions of the form f(x) = a sin[k(x - d)] + c and f(x) = a cos[k(x - d)] + c  The transformation of a sine or cosine function f(x) to g(x) has the general form g(x) = a f [k(x - d)] + c, where |a| is the amplitude, d is the phase shift, and c is the vertical translation.  The period of the transformed function is given by 2π / k.  The k value of the function is given by 2π / period.  Solving Trigonometric Functions  Trigonometric equations can be solved algebraically by hand or graphically using technology.  There are often multiple solutions. Ensure that you find all solutions that lie in the domain of interest.  Quadratic trigonometric equations can often be solved by factoring.  Often, a trigonometric equation might need to be manipulated using trigonometric identities in order of it to be solved. Refer to notes on trigonometric identities here. Unit 6: Logarithmic Functions  Exponential Functions  an exponential function of the form y = bx , b > 0, b not equal 1, has  a repeating pattern of finite differences  a rate of change that is increasing proportional to the function for b > 1  a rate of change that is decreasing proportional to the function for 0 < b < 1  An exponential function of the form y = bx , b > 0, b not equal 1,  has a domain X E R  a range Y E R, Y > 0  a y-intercept of 1  has a horizontal asymptote at y = 0  is increasing on its domain when b > 1  is decreasing on its domain when 0 < b < 1  The inverse of y = bx is a function that can be written as x = by.  has a domain of X E R, x > 0  a range of Y E R  a x-intercept of 1  has vertical asymptote at x = 0  is a reflection of y = bx about the line y = x  is increasing on its domain when b > 1  is decreasing on its domain when 0 < b < 1  Logarithms  a logarithmic function is the inverse of the exponential function  The value of logbx is equal to the exponent to which the base, b, is raised to produce product x  Exponnetial equations can be written in logarithmic form, and vice versa  y = b^x x = logby  y = logbX x = b^y  Exponential and logarithmic functions are defined only for positive values of the base that are not equal to one. In other words, b not = 1, and x > 0.  The logarithm of x to base 1 is only valid when x = 1, in which base y has an infinite number of solutions and is not a function.  Common logarithms are logarithms wit ha base of 10. It is not necessary to write the base for common logarithms: logx means l
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