5-1 Study Guide and Intervention - MRS. FRUGE
[Pages:15]NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-1 Study Guide and Intervention
Trigonometric Identities
Basic Trigonometric Identities An equation is an identity if the left side is equal to the right side for all values of the variable for which both sides are defined. Trigonometric identities are identities that involve trigonometric functions.
Reciprocal Identities
Pythagorean Identities
sin = 1
csc
csc = 1
sin
cos = 1
sec
sec = 1
cos
tan = 1
cot
cot = 1
tan
sin2 + cos2 = 1 tan2 + 1 = sec2 cot2 + 1 = csc2
Example:
If
sin
=
and
0?
<
<
90?,
find
tan
.
Use two identities to relate sin and tan .
sin2 + cos2 = 1
(3)2
5
+
cos2
=
1
cos2
=
16 25
Pythagorean Identity sin = 3
5
Simplify.
cos
=
?
16
25
or
?
4 5
Take the square root of each side.
Since 0? < < 90?, cos is positive. Thus, cos = 45.
Now find tan .
tan
=
sin cos
3
tan
=
5 4
5
tan
=
3 4
Quotient identity
sin = 3, cos = 4
5
5
Simplify.
Exercises
Find the value of each expression using the given information.
1. If cot = 12, find tan .
5
2. If sin = ? 1, find csc . ?4 4
3. If tan = 23, find cot .
4. If sec = ?2, find csc ( - 2). 2
5.
If
cot
=
?
4 3
and
sin
<
0,
find
cos
and
csc
.
cos = , and csc = ?
6. If sec = ?4 and csc > 0, find cos and tan .
cos
=
?
,
and
tan
=
?
Chapter 5
5
Glencoe Precalculus
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-1 Study Guide and Intervention (continued)
Trigonometric Identities
Simplify and Rewrite Trigonometric Expressions You can apply trigonometric identities and algebraic techniques such as substitution, factoring, and simplifying fractions to simplify and rewrite trigonometric expressions.
Example: Simplify each expression.
a. sec x ? cos x
sec
x
-
cos
x
=
1 cos
-
cos
x
= 1 - cos2
cos
= sin2
cos
=
sin
x
(sin
cos
)
= sin x tan x
b.
csc
x
x
+
csc
x
cot2
x
+
1 sin
=
csc
x
cot2
x
+
csc
x
= csc x (csc2 x ? 1) + csc x
= csc3 x ? csc x + csc x
= csc3 x
Reciprocal Identity Add. Pythagorean Identity Factor. Quotient Identity
Reciprocal Identity Pythagorean Identity Distributive Property Simplify.
Exercises Simplify each expression.
1. cos x (tan x + cot x) csc x
2. sin x + cos x cot x csc x
3.
1
csc2 + tan2
x
5. (cot2 x + 1)(sec2 x ? 1) x
7. csc x sin x + cot2 x x
4. (sec x ? tan x)(csc x + 1) cot x
6.
1
+
tan2 1 + sec
sec x
8. cos x (1 + tan2 x ) sec x
9. cos (2 ? ) x csc
10. cos (2 ? ) + cos2 x 1 csc
Chapter 5
6
Glencoe Precalculus
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-1 Practice
Trigonometric Identities
Find the value of each expression using the given information.
1. If cos
=
1 4
and
0?
<
<
90?,
find
tan
.
2.
If sin
=
2 3
and
0?
<
<
90?,
find cos
.
3.
If tan =
7 and 0? <
2
< 90?, find sin .
4. If tan = 2 and 0? < < 90?, find cot .
5. If sin =
?
4 5
and
cos
>
0,
find
tan
and
sec
.
tan = ? , sec =
6. If cot x =
?
3 2
and
sec
x
<
0,
find
sin
x
and
cos
x.
sin x = , cos x = ?
7. If cos = 0.54, find sin ( - 2). ?0.54
8. If cot x = ?0.18, find tan ( - 2). 0.18
Simplify each expression. 9. cos x + sin x tan x sec x
11. sin2 cos2 ? cos2 ?
10.
cot tan
? A
12.
csc2 - cot2 sin (-) cot
?sec x
13. KITE FLYING Brett and Tara are flying a kite. When the string is tied to the ground, the height of the kite can be determined by the formula = csc , where L is the length of the string and is the angle between the string and the
level ground. What formula could Brett and Tara use to find the height of the kite if they know the value of sin ?
H = L sin
Chapter 5
7
Glencoe Precalculus
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-2 Study Guide and Intervention
Verifying Trigonometric Identities
Verify Trigonometric Identities 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.
Example: Verify that
?
=
x.
The left-hand side of this identity is more complicated, so start with that expression first.
sec2 sec2
?
1
=
(tan2 + sec2
1)
-
1
=
tan2 sec2
= (csoins22 ) 1 cos2
=
sin2 cos2
?
cos2
x
= sin2 x
Pythagorean Identity Simplify.
Quotient Identity and Reciprocal Identity Simplify. Multiply.
Notice that the verification ends with the expression on the other side of the identity.
Exercises Verify each identity.
1. sec ? cos = sin tan
sec - cos = ? cos = ? = = sin ( )
= sin tan
2. sec = sin (tan + cot )
sin (tan + cot ) = sin ( + ) = sin ( + )
= sin ( )= = sec
3. tan csc cos = 1 tan csc cos = ( ) ( ) cos = 1
4.
csc2 - cot2 1 - sin2
=
sec2
- ?
=
(
+ ) ?
=
=
Chapter 5
10
Glencoe Precalculus
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-2 Study Guide and Intervention (continued)
Verifying Trigonometric Identities
Identifying Identities and Nonidentities 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.
Example: Use a graphing calculator to test whether csc ? sin = cot cos 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.
csc
?
sin
=
1 sin
?
sin
=
1
- sin2 sin
Rewrite in terms of sine using a Reciprocal Identity. Rewrite using a common denominator.
=
cos2 sin
Pythagorean Identity
=
cos sin
?
cos
Factor cos2 .
= cot cos Rewrite in terms of cot using a Quotient Identity.
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
sin x + cos x cot x
= sin x + cos x ( )
= sin x + = +
= = csc x
2. 2 ? cos2 x = sin2 x
When
x
=
,
=
1.5
and = 0.5;
therefore, the equation is not an identity.
Chapter 5
11
Glencoe Precalculus
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-2 Practice
Verifying Trigonometric Identities
Verify each identity.
1.
cot
csc +
tan
=
cos
x
2.
sin
1
-
1
-
sin
1
+
1
=
-2
sec2
y
=
+
?
+
=
+
-
?
+
=
+ ? + ?
=
?
= =
3. sin3 x ? cos3 x = (1 + sin x cos x)(sin x ? cos x) x ? x = (sin x ? cos x)
( x + sin x cos x + x)
= ?2 y
4.
tan
+
cos 1 + sin
=
sec
tan + = +
+ +
= + +
( )(+ )
= (sin x ? cos x)(1 + sin x cos x)
=
+ ( )(+
)
=
sec
= (1 + sin x cos x)(sin x ? cos x)
5. (sec
-
tan )2 =
1 - sin 1 + sin
6.
1
sin + cos
+
1
+ cos sin
=
2
csc
( ? ) = ? 2 sec tan +
=
?
2(
)
(
)
+
+ + = ++ +
+
( + )
=
?
+
= +
( + )
=
( ? ) ?
=
( ? ) ( + )( ?
)
=
? +
= ( + )
( + )
= = 2 csc
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.
7.
cos 1- sin
=
1 + sin cos
= ? +
? ? +
=
( + ?
)
=
( +
)
= +
8. sin x(sec x + cot x) = cos x
When
x
=
,
=
1.7
and
=
0.7;
therefore, the equation is not an identity.
9. PHYSICS The work done in moving an object is given by the formula W = Fd cos , where d is the displacement, F is
the
force
exerted,
and
is
the
angle
between
the
displacement
and
the
force.
Verify
that
W
=
cot csc
is
an
equivalent
formula. W =
=
( )
=
?
sin
=
Fd
cos
Chapter 5
12
Glencoe Precalculus
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-3 Study Guide and Intervention
Solving Trigonometric Equations
Use Algebraic Techniques to Solve To solve a trigonometric equation, you may need to apply algebraic methods. These methods include isolating the trigonometric expression, taking the square root of each side, factoring and applying the Zero-Product Property, applying the quadratic formula, or rewriting using a single trigonometric function. In this lesson, we will consider conditional trigonometric equations, or equations that may be true for certain values of the variable but false for others.
Example 1: Find all solutions of tan x cos x ? cos x = 0 on the interval [0, 2).
tan x cos x ? cos x = 0
Original equation
cos x (tan x ? 1) = 0
Factor.
cos x = 0 or
tan x ? 1 = 0
Set each factor equal to 0.
x
=
2
or
3 2
tan x = 1
x
=
4
or
5 4
When
x
=
2
or
32,
tan
x
is
undefined,
so
the
solutions
of
the
original
equation
are
4
or
54.
When
you
solve
for
all
values
of
x, the solution should be represented as x + 2n for sin x and cos x and x + n for tan x, where n is any integer. The
solutions
are
4
+
n
or
5 4
+
n.
Example 2: Find all solutions of sin x + = ?sin x.
sin x + 3 = ?sin x
Original equation
2 sin x + 3 = 0
Add sin x to each side.
2 sin x = ? 3
Subtract 3 from each side.
sin
x
=
?
3 2
Divide each side by 2.
x
=
4 3
or
5 3
Solve for x.
The
solutions
are
4 3
+
2n
or
5 3
+
2n.
Exercises Solve each equation for all values of x. 1. cos x = ?1 + 2n
2. sin3 x ? 4 sin x = 0 n
3. sin x cos x ?3 cos x = 0 + n
4. 2 sin3 x = sin x n, + n, + n
Find all solutions of each equation on the interval [0, 2).
5. 2 cos x = 1 ,
6. 5 + 2 sin x ? 7 = 0
7. 4 sin2 x tan x = tan x
8. 2 cos x ? 3 = 0
0, , , , ,
Chapter 5
,
16
Glencoe Precalculus
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
5-3 Study Guide and Intervention (continued)
Solving Trigonometric Equations
Use Trigonometric Identities to Solve You can use trigonometric identities along with algebraic methods to solve trigonometric equations. Be careful to check all solutions in the original equation to make sure they are valid solutions.
Example 1: Find all solutions of 2 x ? x + 3 = 1 ? 2 tan x on the interval [0, 2).
2 tan2 x ? sec2 x + 3 = 1 ? 2 tan x
Original equation
2 tan2 x ? (tan2 x + 1) + 3 = 1 ? 2 tan x tan2 x + 2 = 1 ? 2 tan x
sec2 x = tan2 x + 1 Simplify.
tan2 x + 2 tan x + 1 = 0
Simplify.
(tan + 1)2 = 0
Factor.
tan x = ?1
Take the square root of each side.
x
=
3 4
or
7 4
Solve for x on [0, 2).
Example 2: Find all solutions of 1 + cos x = sin x on the interval [0, 2).
1 + cos x = sin x
Original equation
(1 + cos )2 = (sin x)2 1 + 2 cos x + cos2 x = sin2 x 1 + 2 cos x + cos2 x = 1 ? cos2 x
Square each side. Multiply. Pythagorean Identity
2 cos2 x + 2 cos x = 0
Simplify.
2 cos x (cos x + 1) = 0
Factor.
cos x = 0 or cos x = ?1
x
=
2,
,
3 2
Zero Product Property Solve for x on [0, 2).
Exercises
Solve each equation for all values of x.
1. tan2 x = 1 +
3. sin x cos x ?3 cos x = 0 + n
2. 2 sin2 x ? cos x = 1
+ 2n, + 2n, + 2n
4. cos2 x + sin x + 1 = 0 + 2n
Find all solutions of each equation on the interval [0, 2).
5. cos x = sin x ,
6. 3 cos x tan x ? cos x = 0
,
7. tan2 x + sec x ? 1 = 0
0, ,
8. 1 + cos x = 3 sin x
,
Chapter 5
17
Glencoe Precalculus
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