Download 54Trig_Identities

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:
sin 
Prove that tan  
.
cos 
y
sin  
r
x
cos  
r
y
tan  
x
5.4.2
Trigonometric Identities
Quotient Identities
Reciprocal Identities
Pythagorean Identities
5.4.3
Trigonometric Identities [cont’d]
sinx x sinx =
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
5.4.4
Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify.
a)
cos   sin  tan 
b)
cot 2 
2
1  sin 
5.4.5
Simplifying Trigonometric Expressions
c)
(1 + tan x)2 - 2 sin x sec x
d)
csc x
tan x  cot 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:
3. Look for trigonometric simplifications:
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
1 + sec x
b)
sec x = tan x csc x
secx
c)
tan x sin x + cos x = sec x
secx
5.4.8
Proving an Identity
d)
sin4x - cos4x
= 1 - 2cos2 x
1 - 2cos2x
e)
1
1
2

 2 csc x
1  cos x 1  cos x
2
2 csc x
5.4.9
Proving an Identity
f)
cos A
1  sin A

1  sin A
cos A

2 secA
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)
5.4.11
Using Exact Values to Prove an Identity [cont’d]
.
b) Verify that this statement is true for x =
6
sin x
1  cos x

1  cos x
sin x
Rationalize the
denominator:
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

1  cos x
sin x
1  cos x
sin x
Note the left side of the
equation has the restriction
Therefore,
where n is any integer.
The right side of the
equation has the restriction
Therefore,
And
n is any integer.
, where
5.4.13
Proving an Equation is an Identity
sin2 A  1
1
1
.
Consider the equation
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)
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
2
sin A  sin A

1
1
sin A
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
2
sin A  sin A

1
1
sin A
equation has the restriction:
sin2A - sin A ≠ 0
sin A(sin A - 1) ≠ 0
1
1
sin A
A  0,  or A 

2
Therefore,A  0  2  n or
A   + 2 n, or
L.S. = R.S.

 2  n, where n 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.
A
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