Th
e Foucault pendulum shown in the
photograph provided physical proof in
the middle of the 19th century that Earth
is rotating about its axis. Pendulums can
have many different paths in different
planes. A pendulum’s path is a two-
dimensional curve that you can describe
by
x
- and
y
-displacements from its rest
position as functions of time. You can
predict a pendulum’s position at any
given time using parametric equations.
Pythagorean properties of trigonometric
functions can be used to model periodic
relationships and allow you to conclude
whether the path of a pendulum is an
ellipse or a circle.
Trigonometric Function Properties and
Identities, and Parametric Functions
y
x
1
343
•
Investigate the sum of the squares
of the cosine and sine of the same
argument.
•
Derive algebraically three kinds of
properties expressing relationships
among trigonometric functions.
•
Given a trigonometric expression,
transform it into an equivalent
expression whose form is perhaps
simpler or more useful.
•
Find algebraically or numerically
the solutions to equations involving
circular or trigonometric sines,
cosines, and tangents of one argument.
•
Given equations for a parametric
function, plot the graph and make
conclusions about the geometric figure
that results.
•
Plot graphs of inverse trigonometric
functions and relations.
•
Find exact values of inverse
trigonometric functions.
CHAPTER OBJECTIVES

Overview
Here students broaden their repertoire of trigonometric
properties, adding to the Pythagorean and quotient properties
they learned in Chapter 5. In some cases they are asked to
transform an expression to another form, and at other times they
are asked to prove that a given equation is an identity. Students
reinforce the proof style they learned in geometry, starting with
“Proof:” to show where statement of the identity ends and proving
begins, and ending with a statement of what they have proved,
including the abbreviation “q.e.d.” The properties are also used to
solve equations, adding arcsine and arctangent to the arccosine
learned in Chapter 6. Students also learn to use parametric
functions to plot the inverse circular relation graphs.
Using This Chapter
Th
is chapter falls into the middle of Unit 2. Although some newer
textbooks do not cover topics such as transforming trigonometric
expressions and proving trigonometric identities, there are several
reasons to include them. These concepts help students learn
elementary trigonometric properties, give students an opportunity
to sharpen their algebraic skills, and show students how to write
algebraic proofs.
Teaching Resources
Explorations
Exploration 7-2:
Properties of Trigonometric Functions
Exploration 7-3a:
Transforming an Expression
Exploration 7-3b: Trigonometric Transformations
Exploration 7-3c:
Trigonometric Identities
Exploration 7-4a:
Arccosine, Arcsine, and Arctangent
Exploration 7-4b:
Trigonometric Equations
Exploration 7-5:
Parametric Function Pendulum Problem
Exploration 7-5a:
Parametric Equations for Ellipses

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