8/8/2019 Chapter 4 Precal
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4-1: Introduction to the Pythagorean Property
j Pythagorean Property:
y
4-2: Pythagorean, Reciprocal, and Quotient Properties
j Reciprocal Properties:
y y y
j The Quotient Properties
y y
j The Three Pythagorean Properties
y y y
4-3: Identities and Algebraic Transformation of Expressions
j Transforming Trigonometric Expressions and Proving
Identities1. Start by writing the given expression or, for an
identity, by
picking the side of the equation you wish to start with
andwriting it down. Usually, it is easier to start with the
morecomplicated side.
2. Look for algebraic things to do.a. If there are two terms and
you only want one term, then
i. Add fractions, or ii. Factor something out.
b. Multiply be a clever form of 1 in order toi. Multiply a
numerator or denominator by its conjugate
binomial.ii. Get a desired expression into the numerator or
denominator.c. Perform any obvious calculations (distribute,
square,
multiply polynomials, and so on).d. Factor out an expression you
want to appear in the result.
3. Look for trigonometric things to do.a. Look for familiar
trigonometric expressions you can
transform.b. If there are squares of functions, think of
Pythagorean
properties.c. Reduce the number of different functions,
transforming
them into the ones you want in the result.d. Leave any unchanged
expressions you want in the answer
4. Keep looking at the result and thinking of ways you can
getcloser to it.
4-4: Arcsine, Arctangent, Arccosine, and
TrigonometricEquations
j Remember that arcos x means any of the angles whose cosineis
x. The same goes for sine and tangent.
j Within any one revolution there are two values of the
inversetrigonometric relation for any given argument.
j When youre asked to solve find solutions to equations
involvingcircular or trigonometric sines, cosines, and tangents of
oneargument, you can always graph it on your calculator and
solve
from there.j Interval Notation:
written meaning
j General Solutions for Arcsine, Arccosine, and Arctangenty
y
y
4-5: Parametric Functions
j Parametric equations: The two equations that express
thecoordinates x and y of points on the curve as separate
functionsof a common variable (the parameter).
j Ellipse: The set of all points P in a plane for which the sum
of the distances from point P to two fixed points is constant.
j Parametric Equations for an Ellipse:y y y Where a and b are
called the x- and y-radii, respectively, and h and k are the
coordinates of the center. If a = b, the figure is a circle.
4-6: Inverse Trigonometric Relation Graphs
j Criteria for Selecting Principal Branches of
InverseTrigonometric Functions:1. The principal branch must be a
function.2. It must use the entire domain of the inverse
trigonometric relation.3. It should be one continuous graph, if
possible.4. It should be centrally located, near the origin.5. If
there is a choice between two possible branches, use the
positive
one.j Ranges and Domains of Inverse Trigonometric Functions:
FUNCTION RANGE(numerically) RANGE(graphically) DOMAIN
Quadrants I & IV Quadrants I & II Quadrants I & IV
Quadrants I & II Quadrants I & II Quadrants I & IV
j The Composite of a Function and Its Inverse Function:y y
provided x is in the range of the outside function and in the
domain of the inside function.
