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Chapter 5 Trigonometric Equations
5.4
MATHPOWERTM 12, WESTERN EDITION 5.4.1
Trigonometric Identities
A trigonometric equation is an equation that involves
at least one trigonometric function of a variable. The
equation is a trigonometric identity if it is true for all
values of the variable for which both sides of the
equation are defined.
Recall the basic
trig identities:
y
sin  
r
x
cos  
r
y
tan  
x
sin 
Prove that tan  
.
cos 
y
y x

 
x
r r
y r
 
r x
y

x
L.S. = R.S.
5.4.2
Trigonometric Identities
Quotient Identities
cos 
cot  
sin 
sin 
tan  
cos 
Reciprocal Identities
1
sin  
csc 
1
cos  
sec 
1
tan  
cot 
Pythagorean Identities
sin2 + cos2 = 1
tan2 + 1 = sec2
cot2 + 1 = csc2
sin2 = 1 - cos2
cos2 = 1 - sin2
tan2 = sec2 - 1
cot2 = csc2 - 1
5.4.3
Trigonometric Identities [cont’d]
sinx x sinx = sin2x
1
cos 2 
1
cos 2   1
cos  



cos  cos  cos 
cos 
sin A  cos A  sin2 A  2sin Acos A  cos 2 A
2
 1  2sin Acos A
cos A
sin A  cos A  sin A
1
sin A
1
sin A
= cosA
5.4.4
Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify.
a)
cos   sin  tan 
sin
 cos   sin 
cos 
sin2 
 cos  
cos 
cos   sin 

cos 
1

cos 
2
 sec 
2
cot 2 
1  sin2 
b)
cos 2 
2
sin


cos 2 
1
cos 2 
1


2
sin  cos2 

1
2
sin 
 csc 2 
5.4.5
Simplifying Trigonometric Expressions
c)
(1 + tan x)2 - 2 sin x sec x
1
cos x
sin x
2
 1  2 tan x  tan x  2
cos x
 (1  tan x)  2 sin x
2
 1  tan2 x  2tanx  2 tanx
 sec2 x
csc x
d) tan x  cot x
1

sin x
sin x cos x

cos x sin x
1

sin x
sin 2 x  cos 2 x
sin xcos x
1

sin x
1
sin x cos x
1
sin x cos x


sin x
1
 cos x
5.4.6
Proving an Identity
Steps in Proving Identities
1. Start with the more complex side of the identity and work
with it exclusively to transform the expression into the
simpler side of the identity.
2. Look for algebraic simplifications:
• Do any multiplying , factoring, or squaring which is
obvious in the expression.
• Reduce two terms to one, either add two terms or
factor so that you may reduce.
3. Look for trigonometric simplifications:
• Look for familiar trig relationships.
• If the expression contains squared terms, think
of the Pythagorean Identities.
• Transform each term to sine or cosine, if the
expression cannot be simplified easily using other ratios.
4. Keep the simpler side of the identity in mind.
5.4.7
Proving an Identity
Prove the following:
a) sec x(1 + cos x) = 1 + sec x
= sec x + sec x cos x
= sec x + 1
1 + sec x
L.S. = R.S.
b)
sec x = tan x csc x
secx
sin x
1

cos x sin x
1

cos x

 secx
L.S. = R.S.
c)

tan x sin x + cos x = sec x
secx
sin x sin x

 cos x
cos x
1
sin2 x  cos 2 x

cos x
1

cos x
L.S. = R.S.
 secx
5.4.8
Proving an Identity
sin4x - cos4x
d)
= 1 - 2cos2 x
= (sin2x - cos2x)(sin2x + cos2x) 1 - 2cos2x
= (1 - cos2x - cos2x)
= 1 - 2cos2x
L.S. = R.S.
1
1
2
e)

 2 cs c x
1  cos x 1  cos x
(1  cos x)  (1  cos x)

2 csc2 x
(1  cos x)(1  cos x)
2

2
(1  cos x)
2

2
sin x
 2 csc2 x
L.S. = R.S.
5.4.9
Proving an Identity
f)
cos A
1  sin A

1  sin A
cos A

cos 2 A (1  sinA)(1  sin A)

(1  sin A)(cos A)
2 secA
2 secA
cos 2 A (1  2sin A  sin2 A)

(1  sinA)(cos A)
cos 2 A  sin2 A 1  2sin A

(1  sinA)(cos A)
2  2sin A

(1  sin A)(cos A)
2(1  sin A)

(1  sin A)(cos A)
2

(cos A)
 2sec A
L.S. = R.S.
5.4.10
Using Exact Values to Prove an Identity
Consider sin x  1  cos x .
1  cos x
sin x
a) Use a graph to verify that the equation is an identity.

b) Verify that this statement is true for x = .
6
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
a)
1 sin
cos
xx
y
1 sin
cos
xx
5.4.11
Using Exact Values to Prove an Identity [cont’d]

b) Verify that this statement is true for x = .
sin x
1  cos x
sin
1  cos x
sin x


6


1  cos
6
1

2
3
1
2
1
2
 
2 2 3
1

2 3
2 3

1  cos
sin
 1


6
Rationalize the
denominator:
1
2 3
6
6
3
2
1
2
2 3 2


2
1
2 3
L.S. = R.S.

1
2 3

2  3 2 3

2 3
43
2 3
Therefore, the identity is
true for the particular

x

.
case of
6
5.4.12
Using Exact Values to Prove an Identity [cont’d]
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
Restrictions:
sin x
1  cos x

Note the left side of the
1  cos x
sin x
equation has the restriction
sin x
1  cos x
1  cos x
1 - cos x ≠ 0 or cos x ≠ 1.


Therefore, x ≠ 0 + 2 n,
1  cos x 1  cos x
sin x
where n is any integer.
sin x(1  cos x)

1  cos x
2
The right side of the
equation has the restriction
sin x ≠ 0. x = 0 and 
Therefore, x ≠ 0 + 2n
and x ≠  + 2n, where
n is any integer.
sin x(1  cos x)

2
sin x
1  cos x

sin x
L.S. = R.S.
5.4.13
Proving an Equation is an Identity
2
sin
A 1
1
Consider the equation
1
.
2
sin A  sin A
sin A
a) Use a graph to verify that the equation is an identity.
b) Verify that this statement is true for x = 2.4 rad.
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
a)
sin21A  1
y  1 2
sin sin
A Asin A
5.4.14
Proving an Equation is an Identity [cont’d]
b) Verify that this statement is true for x = 2.4 rad.
sin2 A  1
sin2 A  sin A

1
1
sin A
(sin 2.4)2  1

2
(sin 2.4)  sin2.4
1
 1
sin 2.4
= 2.480 466
= 2.480 466
L.S. = R.S.
Therefore, the equation is true for x = 2.4 rad.
5.4.15
Proving an Equation is an Identity [cont’d]
c) Use an algebraic approach to prove that the identity is true
in general. State any restrictions.
Note the left side of the
sin2 A  1
sin2 A  sin A

1
1
sin A
(sin A  1)(sin A  1)

sin A(sin A  1)
1
1
sin A
(sin A  1)

sin A
sin A
1


sin A sin A
1
 1
sin A
L.S. = R.S.
equation has the restriction:
sin2A - sin A ≠ 0
sin A(sin A - 1) ≠ 0
sin A ≠ 0 or sin A ≠ 1

A  0,  or A 
2
Therefore,A  0  2  n or
A   + 2 n, or

A   2  n, wheren is
2
any integer.
The right side of the
equation has the restriction
sin A ≠ 0, or A ≠ 0.
Therefore, A ≠ 0,  + 2 n,
where n is any integer.
5.4.16
Suggested Questions:
Pages 264 and 265
A 1-10, 21-25, 37,
11, 13, 16
B 12, 20, 26-34
5.4.16