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Math 152-copyright Joe Kahlig, 11C
Review of Functions and Appendix D
Functions: A function, f , is a rule that assigns to each element, x, in set A exactly one element,
called f (x), in set B. The set A is called the domain. The range of f is the set of all possible values
of f (x) where x is in the domain, i.e. range = { f (x)|x ∈ A }.
Example: Find the domain of f (x) =
Example: Find the domain of y = √
x
x2 − 25
x2
1
− 2x − 24
Example: Express y = |x − 5| without the absolute value symbols.
Example: Express y = |x2 − 4| without the absolute value symbols.
Appendix D: Trigonometry
Angles can be measured in degrees or radians. Thus 360o = 2π radians
Example: Convert 250 to radians.
Example: Convert
−3π
to degrees.
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Math 152-copyright Joe Kahlig, 11C
Trigonometric Functions: Consider the right triangle.
hypotenuse
sin A =
opp
hyp
cos A =
adj
hyp
tan A =
opp
adj
csc A =
hyp
opp
sec A =
hyp
adj
cot A =
adj
opp
opposite
A
adjacent
Special Triangles: 45-45-90 and 30-60-90.
Example: Evaluate the following:
2π
sin
3
=
Unit Circle:
2π
cos
3
=
2π
tan
3
=
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Math 152-copyright Joe Kahlig, 11C
Trigonometric Identities: These identities are used some in Math 151, but are use a lot in Math 152.
sin θ
cos θ
sin2 θ + cos2 θ = 1
sin(2x) = 2 sin x cos x
tan2 θ + 1 = sec2 θ
tan θ =
1 + cos(2x)
2
1
−
cos(2x)
sin2 x =
2
cos2 x =
cos(2x) = 2 cos2 (x) − 1
Law of sines:
B
sin A
sin B
sin C
=
=
a
b
c
a
c
Law of cosines: a2 = b2 + c2 − 2bc cos A
C
A
b
Example: If sec(x) =
Law of cosines: b2 = a2 + c2 − 2ac cos B
Law of cosines: c2 = a2 + b2 − 2ab cos C
−π
3
with
< x < 0, find sin(2x)
2
2
Example: Solve the following equations for x where 0 ≤ x ≤ 2π
A) 2 cos(x) − 1 = 0
B) 2 cos(x) + sin(2x) = 0