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Amy
Austin, August 23, 2013
Brief review of Algebra Concepts
Functions
EXAMPLE 1: Find the domain of f (x) =
EXAMPLE 2: Find the domain of f (x) =
EXAMPLE 3: Given f (x) =
√
x2
x+1
− 2x − 3
√
√
8−x+
x + 5, find and simplify
x2 − 4
f (4 + h) − f (4)
h
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Amy
Austin, August 23, 2013
EXAMPLE 4: Given f (x) =
4
f (x + h) − f (x)
, find and simplify
x+2
h
Equations of Lines
EXAMPLE 5: Find the equation of the line that passes through the points A(−1, 3)
and B(2, 5). Find the x and y intercept of this line. Find the equation of the
perpendicular bisector of AB. Find the equation of the circle for which AB is a
diameter.
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Amy
Austin, August 23, 2013
Appendix D: Review of Trigonometry
Measurement of Angles Angles can be measured in degrees or radians. The angle
given by a complete revolution contains 360◦ , or 2π radians.
Thus, 360◦ = 2π radians, yielding the conversion formulas
180
π
◦
= 1 radian
π
radians = 1◦
180
EXAMPLE 6:
a.) Convert 36◦ to radians.
b.) Convert −
3π
to degrees.
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The Six Trigonometric Functions Consider the right triangle below:
We define
opposite
sin A =
hypotenuse
opposite
tan A =
adjacent
hypotenuse
sec A =
adjacent
adjacent
hypotenuse
adjacent
cot A =
opposite
hypotenuse
csc A =
opposite
cos A =
12
and x is a quadrant IV angle, find the value of sin(x),
13
tan(x), csc(x), sec(x), and cot(x)
EXAMPLE 7: If cos x =
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Amy
Austin, August 23, 2013
Pythagorean Theorem and two special triangles
The Unit Circle
If we are using the unit circle to find the trig ratios for a given angle θ, then
1
x
1
y
cos(θ) = x, sin(θ) = y, tan(θ) = , sec(θ) = , cot(θ) = and csc(θ) =
x
x
y
y
5π
EXAMPLE 8: Find the exact trig ratios for θ =
.
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4
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Amy
Austin, August 23, 2013
Identities to recall
sin θ
tan θ =
cos θ
cos θ
cot θ =
sin θ
2
sin θ + cos2 θ = 1
1
cos θ
1
csc θ =
sin θ
2
tan θ + 1 = sec2 θ
sec θ =
sin(2θ) = 2 sin θ cos θ
cos(2θ) = 2 cos2 θ − 1
1
1
cos2 θ = (1 + cos(2θ))
sin2 θ = (1 − cos(2θ))
2
2
EXAMPLE 9: Solve the following equations for x, where 0 ≤ x ≤ 2π.
a.) 2 cos2 x − 1 = 0
b.) 2 cos x + sin 2x = 0
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Amy
Austin, August 23, 2013
5
π
EXAMPLE 10: If sec(x) = , − < x < 0, find sin(2x).
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2
Graphs of sin x, cos x, tan x
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