Download Holt Algebra 2 Date Dear Family, In Chapter 14 your child will

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Dear Family,
In Chapter 14 your child will continue to study trigonometry, specifically
graphing trigonometric functions, using trigonometric identities, and solving
trigonometric equations.
The graphs of the basic trigonometric functions are all periodic functions
that repeat in regular intervals called cycles. The length of the cycle is
called the period of the function. The amplitude of the function is half the
distance between the maximum and minimum values of the function. Here
are the basic shapes of the trigonometric functions:
Graphs of Trigonometric Functions
Sine
Cosine
Tangent
y sin x
y cos x
y tan x
period: 2
amplitude: 1
period: amplitude: undefined
period: 2
amplitude: 1
Y
Y
R
R
R
X
R
R
R
R
X
R
R
R
Cosecant
Secant
Cotangent
y csc x
y sec x
y cot x
Y
Y
R
R
X
.
.
X
R
period: amplitude: undefined
period: 2
amplitude: undefined
Y
R
R
period: 2
amplitude: undefined
R
Y
.
.
X
R
R
R
R
X
Many real-world phenomena, such as sound, occur in periodic waves that
can be modeled by a sine or cosine function. In these situations, it is often
useful to describe the number of cycles that occur in a given unit of time,
called the frequency of the function.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
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Holt Algebra 2
As with all of the functions that your child has
studied in Algebra 2, trigonometric functions
can be transformed by stretching, compressing,
and translating the graph. Vertical stretches and
compressions will affect the amplitude of the
function. Horizontal stretches and compressions
affect the period and vertical asymptotes (if any).
For trigonometric functions, a horizontal translation
is also called a phase shift.
Y
R
R
Compared to y sin x, the graph of y 3 sin x __
2
is stretched vertically by a factor of 3 (changing the
amplitude from 1 to 3), and is horizontally translated right
.
by a phase shift of __
2
R
R
X
An identity is a mathematical statement that is true for all values of the
variable for which the statement is defined. For example, the statement
2
2
cos sin 1 is true for any angle , and is therefore a trigonometric
identity. (Specifically, the previous identity is one of several based on the
Pythagorean Theorem, and hence called a Pythagorean Identity.)
In Chapter 14, your child will learn over two dozen trigonometric identities,
and prove several more. Here’s one application of these identities:
Find the exact value of sin 15⬚.
sin 15 sin 45 30 sin 45 cos 30 cos 45 sin 30
3 ___
2 ___
2 __
1
___
2
2
2 2
6 2
6 ___
2 ________
___
4
4
4
Write 15 as a sum or difference of
familiar angles from the unit circle.
One of the Difference Identities is
sin (A B) sin A cos B cos A sin B.
Evaluate. (Refer to the unit circle in
the Parent Letter for Chapter 13.)
Simplify.
The chapter ends with a lesson on solving trigonometric equations.
Solving these types of equations calls upon many of the skills your
child has learned previously, including trigonometric inverses, graphs
of trigonometric functions, trigonometric identities, and even quadratic
equations and factoring.
Many situations that involve periodic occurrences, circular motion, or
angles, can be modeled by trigonometric functions. Sound waves, the
motion of a Ferris wheel, the depth of tides, the time that the sun rises and
sets, and the forces acting upon a car parked on a hill can all be analyzed
with trigonometry.
For additional resources, visit go.hrw.com and enter the keyword MB7 Parent.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
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Holt Algebra 2