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DATE
5-2
PERIOD
NAME
DATE
5-2
Study Guide and Intervention
Verifying Trigonometric Identities
You can use a graphing
calculator to test whether an equation might be an identity by graphing the
functions related to each side of the equation. If the graphs of the related
functions do not coincide for all values of x for which both functions are
defined, the equation is not an identity. If the graphs appear to coincide, you
can verify that the equation is an identity by using trigonometric properties
and algebraic techniques.
sec2 x − 1
= sin2 x.
Verify that −
2
sec x
The left-hand side of this identity is more complicated, so start with that
expression first.
2
sec x
(tan2 x + 1) − 1
sec x
tan2 x
=−
2
sec x
(
Simplify.
)
Quotient Identity and Reciprocal Identity
sin θ
1 - sin2 θ
=−
sin x
=−
· cos2 x
2
2
sin θ
cos2 θ
=−
sin θ
cos θ
=−
· cos θ
sin θ
Simplify.
cos x
= sin x #
Multiply.
Notice that the verification ends with the expression on the other side of the
identity.
2
1. sec θ - cos θ = sin θ tan θ
( cos θ )
1 − cos2 θ
sin2 θ
sin θ
1
sec θ − cos θ = −
− cos θ = −
=−
= sin θ −
cos θ
cos θ
cos θ
= sin θ tan θ
2. sec θ = sin θ(tan θ + cot θ)
( cos θ
)
cos θ
sin θ
sin θ (tan θ + cot θ) = sin θ −
+−
= sin θ
( cos θ sin θ )
1
= sin θ −
1
=−
= sec θ
sin θ
(
sin2 θ + cos2 θ
−
cos θ sin θ
)
cos θ
3. tan θ csc θ cos θ = 1
( cos θ )( sin θ )
sin θ
1
−
tan θ csc θ cos θ = −
cos θ = 1
θ − cot θ
−
= sec2 θ
4. csc
2
2
2
Chapter 5
005-032_PCCRMC05_893806.indd 10
10
[-2π, 2π] scl: π by [-4, 4] scl: 1
2
Pythagorean Identity
Factor cos2 θ.
Rewrite in terms of cot θ using a Quotient Identity.
[-2π, 2π] scl: π by [-4, 4] scl: 1
2
Exercises
Test whether each equation is an identity by graphing. If it appears
to be an identity, verify it. If not, find an x-value for which both
sides are defined but not equal.
1. sin x + cos x cot x = csc x
2. 2 - cos2 x = sin2 x
π
, Y1 = 1.5
When x = −
sin x + cos x cot x
4
( sin x )
cos x
= sin x + cos x −
and Y2 = 0.5;
therefore, the equation
is not an identity.
sin2 x + cos2 x
sin x
cos x
=−
= sin x + −
2
sin x
1
=−
= csc x
1 − sin θ
(cot2 θ + 1) − cot2 θ
csc2 θ − cot2 θ
1
−
= −−
=−
= sec2 θ
cos2 θ
cos2 θ
1 − sin2 θ
sin x
Glencoe Precalculus
Chapter 5
3/13/09 12:10:25 PM
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Glencoe Precalculus
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Verify each identity.
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A4
= cot θ cos θ #
Exercises
Rewrite using a common denominator.
005-032_PCCRMC05_893806.indd 11
11
Glencoe Precalculus
3/13/09 12:10:29 PM
Answers (Lesson 5-2)
=
Example
Use a graphing calculator to test whether csc θ - sin θ
= cot θ cos θ is an identity. If it appears to be an identity, verify it. If
not, find an x-value for which both sides are defined but not equal.
The equation appears to be an identity because the graphs
of the related functions coincide. Verify this algebraically.
1
csc θ - sin θ = −
- sin θ Rewrite in terms of sine using a Reciprocal Identity.
Pythagorean Identity
sin2 x
−
cos2 x
−
1
−
cos2 x
(continued)
Identifying Identities and Nonidentities
To verify an identity means to
prove that both sides of the equation are equal for all values of the variable
for which both sides are defined.
sec x − 1
−
= −
2
2
Study Guide and Intervention
Verifying Trigonometric Identities
Verify Trigonometric Identities
Example
PERIOD
Lesson 5-2
Chapter 5
NAME
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
DATE
5-2
PERIOD
NAME
DATE
5-2
Practice
Word Problem Practice
Verifying Trigonometric Identities
Verifying Trigonometric Identities
Verify each identity.
csc x
1. −
= cos x
sin y − 1
1
−
sin x
sin y + 1
sin y + 1 − sin y + 1
csc x
sin x cos x
= − · −
−
cos x
cot x + tan x
sin x
sin x cos x
1
1
− −
= −
−
sin y − 1
sin y + 1
sin2 y − 1
−
+−
cos x
sin x
=
1. SURVEYING From a point h meters
above level ground, a surveyor measures
the angle of depression θ of the corner of
a building lot. The distance d from the
corner to the point on the ground directly
under the surveyor’s instrument can be
described by the equation d = h cot θ.
1
1
2. −
−−
= −2 sec2 y
cot x + tan x
cos x
−
cos2 x + sin2 x
=
cos x
= cos x #
=−
1
2
−
-cos2 y
( 1 - tan θ )
cot θ - 1
Verify that d = h −
is an
cos θ
3. sin3 x - cos3 x = (1 + sin x cos x)(sin x - cos x) 4. tan θ + −
= sec θ
1 + sin θ
)(
)
1 + cos θ
sin2 θ + 1 + 2 cos θ + cos2 θ
sin θ
−
+ − = −−
1 + cos θ
sin θ
sin θ (1 + cos θ)
(1 − sin θ)2
1 − sin θ
= −
2 + 2 cos θ
sin θ (1 + cos θ)
2 (1 + cos θ)
sin θ (1 + cos θ)
= −
2
(1 − sin θ)2
1 − sin θ
= −− = −
1 + sin θ
(1 + sin θ)(1 − sin θ)
2
=−
= 2 csc θ #
#
sin θ
Test whether each equation is an identity by graphing. If it appears
to be an identity, verify it. If not, find an x-value for which both
sides are defined but not equal.
1 + sin x
cos x
cos x
7. −
=−
1 − sin x
8. sin x(sec x + cot x) = cos x
π
When x = −
, Y = 1.7 and Y2 = 0.7;
4 1
1 + sin x
1 − sin x 1 + sin x
1 − sin x
cos x (1 + sin x)
= −
1 − sin2 x
cos x (1 + sin x)
= −
cos2 x
1 + sin x
= −
#
cos x
cos x
cos x
−
=−
·−
therefore, the equation is not an identity.
Glencoe Precalculus
9. PHYSICS The work done in moving an object is given by the formula
W = Fd cos θ, where d is the displacement, F is the force exerted, and θ is
the angle between the displacement and the force. Verify that
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
= −
csc θ
Fd cot θ
−
csc θ
Fd
=
Chapter 5
Answers
005-032_PCCRMC05_893806.indd 12
cos θ
(−
sin θ )
−
1
−
sin θ
=
Fd cos θ
−
sin θ
· sin θ = Fd cos θ
12
5θ − 6(sec θ − tan θ sin θ)
)
( cos θ cos θ
sin θ
1
−−
= 5θ − 6(−
cos θ )
cos θ
1 − sin θ
= 5θ − 6( −
cos θ )
cos θ
= 5θ − 6 (−)
cos θ
sin θ
1
= 5θ − 6 −
−−
· sin θ
2
)
2
= 5θ − 6 cos θ #
2. CAROUSEL The angular velocity ω of
each person riding a carousel is the
number of degrees, or radians, that a
point on the edge of the carousel moves
in a unit of time. You can use the
equation y = r sin (60ω), where r is the
radius of the circle, to sketch a graph
representing the position of a person on
the carousel with respect to a stationary
point every 60 seconds.
-
4. TELESCOPE The largest working
refracting telescope is housed at the
University of Chicago’s Yerkes
Observatory. In a refracting telescope, as
the diameter d of the lens increases, its
resolution θ can be improved, which
means the telescope can better separate
images in the sky. The relationship
between resolution θ in degrees and
diameter d in meters can be expressed by
ω
1.22λ
sin θ = −
, where λ is the wavelength
d
S
of light in meters, or by
csc θ + 1
1.22λ
− =−
. Verify that
1
+ csc θ
(−
sin θ )
a. Verify that the equation
y = r tan (60ω) cos (60ω) is an
equivalent formula.
2
1
+ csc θ
(−
sin θ )
2
r tan (60ω) cos (60ω)
=
r sin (60ω)
−
cos (60ω)
csc θ + 1
3/13/09 12:10:35 PM
005-032_PCCRMC05_893806.indd 13
csc θ + 1
csc θ + csc θ
− = −
2
· cos (60ω)
(
)
1
+ csc θ
−
sin2 θ
= r sin (60ω)
Chapter 5
d
csc θ + 1
sin θ = − is an identity.
csc θ + 1
= −
csc θ (csc θ +1)
1
= sin θ #
=−
csc θ
graphs should appear to coincide.
Glencoe Precalculus
)
2
b. How could you use a graphing
calculator to check your answer
in part a? Graph each function. The
Fd cot θ
W=−
is an equivalent formula.
W=
)
cos θ
(
cos θ
= h cot θ #
= h( −
sin θ )
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
A5
(
cos θ - sin θ
−
cos θ
sin θ
1-−
θ - sin θ
cos θ
−
= h cos
· −
sin θ
cos θ - sin θ
(sec θ - tan θ)2 = sec2 θ - 2 sec θ tan θ + tan2 θ
1 − 2 sin θ + sin2 θ
cos θ
sin θ
=h −
1 + cos θ
sin θ
6. −
+ − = 2 csc θ
1 + cos θ
sin θ
= −−
2
cos θ - sin θ
−
cos θ
−
-1
sin θ
−
13
Glencoe Precalculus
3/13/09 12:10:41 PM
Answers (Lesson 5-2)
(
cos θ
sin θ
cos θ
tan θ + − = − + −
cos θ
1 + sin θ
1 + sin θ
sin θ + sin2 θ + cos2 θ
= −
= (sin x - cos x)(1 + sin x cos x)
(cos θ)(1 + sin θ)
1 + sin θ
= (1 + sin x cos x)(sin x - cos x) #
= − = sec θ #
(cos θ)(1 + sin θ)
sin2 θ
sin θ
+−
−
cos θ
cos2 θ
(
( 1 - tan θ ) = h
cot θ - 1
d=h −
sin3 x - cos3 x = (sin x - cos x)
(sin2 x + sin x cos x + cos2 x)
1
1
=−
−2 −
cos θ
cos2 θ
3. TRAINS The equation y = 5θ - 6 cos θ
can be used to help describe the
movement of a train’s wheel as the train
moves forward. Use a graphing calculator
to test whether the equation 5θ - 6 cos θ
= 5θ - 6 (sec θ -tan θ sin θ) is an
identity. If it appears to be an identity,
verify it. If not, find a value for which
both sides are defined but not equal.
equivalent equation.
= -2 sec2 y #
1 − sin θ
5. (sec θ - tan θ)2 = −
1 + sin θ
PERIOD
Lesson 5-2
Chapter 5
NAME