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Lecture

# unit 5 to 7.docx

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University of Toronto Scarborough

Mathematics

MATA30H3

Ken Butler

Winter

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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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