Download Chapter 4: Identities Section 4.1: Proving Identities Basic Identities

Chapter 4:
Identities
Section 4.1:
Proving Identities
Basic Identities:
cos x  sin x  1
sin x
tan x 
cos x
1
1
sec x 
,cos x 
cos x
sec x
1
1
csc x 
,sin x 
sin x
csc x
1
1
cot x 
, tan x 
tan x
cot x
2
2
Pythagorean Identities:
cos x  sin x  1
sin x  1 cos x
cos x  1  sin x
1+ tan x  sec x
cot x  1  csc x
2
2
2
2
2
2
2
2
2
2
Simplify the following expressions
cos( x)  cos x
sin( x)   sin x
tan( x)   tan x
Steps to Simplify an Expression:
1. Recognize basic identities in the expression. Use to rewrite in
either an equivalent form or in terms of sine and cosine.
2. Perform indicated operations and continue, as needed, using
identities along arithmetic or algebraic skills.
Prove:
sec x ( sec x – cos x) = tan2 x
Prove the following are identities:
is an identity
Steps to prove identities p241.
1. Select one side to work on, usually the more complicated.
2. Use basic identities to rewrite into equivalent forms in terms of
sine and cosine. Look at other side for clues.
3. Perform indicated operations using trig and algebra skills.
Prove the following identities:
Look at Connections with Technology p243
HW 4.1, pp245-246: 1 – 13, 15 – 55 odd.
Section 4.2:
Sum and Difference Identities for Cosine
Cosine Difference:
(cos( A  B)  1)2  (sin( A  B)  0)2 )  (cos A  cos B)2  (sin A  sin B)2
a fair amount of algebra later·······
cos( A  B)  cos A cos B  sin A sin B
cosine difference identity
Cosine Sum:
cos (A + B) = cos (A – (–B))
cos (A + B) = cos A cos (–B) + sin A sin (–B)
cos (A + B) = cos A cos B – sin A sin B
cosine sum identity
Ex, Prove the following identity:


cos   x   sin x
2

Ex, Find the following exact functional values:
cos

12
Ex, Find the exact value of:
cos105
cos 57º cos 12º + sin 57º sin 12º
If cos A 
1
and sin B  5 / 26 with angles A and B in QIV,
2
find cos (A + B)
Find the exact value for cos (A – B) if tan A = ½ and sin B = 3/5
where -90º < A < 90º and -90º < B < 90º
HW 4.2, pp254-258: 1 – 9, 11 – 27 odd, 28 – 31, 33 – 55 odd, 59.
Section 4.3:
Sum and Difference Identities for Sine and Tangent
Sine Sum and Difference Identities:
sin (A + B) = sin A cos B + cos A sin B
sin (A – B) = sin A cos B – cos A sin B
Tangent Sum and Difference Identities:
tan (A  B) 
tan A  tan B
1  tan A tan B
tan (A  B) 
tan A  tan B
1  tan A tan B
Cofunction Identities:

cos(  x)  sin x
2




sin   x  = cos x tan   x  = cot x
2

2

Find the following exact functional values:
Use either the sine or tangent sum/difference identities to write each of
the following expressions as a single function of x.
If sin A = 3/5 with A in QIII and cos B = 2/3 in QIV, find the exact value
and then approximate the value rounded to four decimal places.
HW 4.3, pp263-265: 1 – 9, 11 – 25 odd, 26, 27, 28, 35 – 49 odd.
Section 4.4:
Double-Angle Identities
Double-Angle Identities (Formulas)
sin 2 A  2sin A cos A
cos 2 A  cos A  sin A
2
2
cos 2A = 1 – 2sin2 A
cos 2A = 2cos2 A – 1
If sin x = -4/5 with x in QIII, find the following:
If cos 2x = -12/13 and 2x is in QIII, find cos x.
tan 2 A 
2 tan A
1  tan A
2
Use the double-angle identities to rewrite each expression as a single
function of a multiple angle.
Rewrite each expression as a trigonometric function of x.
HW 4.4, pp270-272: 1 – 11, 13 – 41 odd.
Section 4.5:
Half-angle and Additional Identities
Half-Angle Identities (Formulas)
sin
2
cos
2
x 1  cos x
x
1  cos x

 sin = 
2
2
2
2
x 1  cos x
x
1  cos x

 cos = 
2
2
2
2
x
1  cos x
sin x
1  cos x
tan  

=
2
1  cos x 1  cos x
sin x
Use a half-angle identity to find the exact values:
Using Half-Angle Identities to Prove Identities/Simplify Expressions
Prove the following identities:
Product-to-Sum Identities (Formulas)
1
cos( A  B)  cos( A  B)
2
1
sin A sin B   cos( A  B)  cos( A  B) 
2
1
sin A cos B  sin( A  B)  sin( A  B) 
2
1
cos A sin B  sin( A  B)  sin( A  B) 
2
cos A cos B 
Write cos 2x sin 8x as a sum or difference
Sum-to-Product Identities (Formulas)
 A B  A B 
cos A  cos B  2cos 
 cos 

 2   2 
 A B  A B 
cos A  cos B  2sin 
 sin 

2

  2 
 A B  A B 
sin A  sin B  2sin 
 cos 

 2   2 
 A B  A B 
sin A  sin B  2sin 
 cos 

2

  2 
Prove:
 A B  A B 
cos A  cos B  2cos 
 cos 

2

  2 
HW 4.5, pp280-226: 1 – 9 odd, 11 – 21, 23 – 65 odd, 71.