Ch 5 Notes F07 O’Brien
LHS Trig 8th ed
Trigonometry
Chapter 5 Lecture Notes
Section 5.1
I.
Fundamental Identities
Negative-Angle Identities
sin (– θ) = – sin θ
csc (– θ) = – csc θ
tan (– θ) = – tan θ
cot (– θ) = – cot θ
cos (– θ) = cos θ
sec (– θ) = sec θ
One of the easiest ways to remember the negative-angle identities is to remember that only cosine
and its reciprocal, secant are even functions. For even functions, f(– x) = f(x) which means these
functions have y-axis symmetry. The other four trig functions (sine, cosecant, tangent, and
cotangent) are odd functions. For odd functions, f(– x) = – f(x) which means these functions have
origin symmetry.
Example 1 Since tangent is an odd function, if tan x = 2.6, then tan (–x) = –2.6. (#1)
II.
Reciprocal Identities
csc θ 
III.
1
sin θ
sec θ 
1
cos θ
cot θ 
cos θ
sin θ
cot θ 
1
tan θ
Quotient Identities
tan θ 
sin θ
cos θ
IV. Pythagorean Identities
sin 2 θ  cos 2 θ  1
V.
tan 2 θ  1  sec 2 θ
1  cot 2 θ  csc 2 θ
Using the Fundamental Identities
A.
Finding Trigonometric Function Values Given One Value and the Quadrant
Example 2 Given cot x  
1
and x is in quadrant IV, find sin x. (modified #6)
3
csc 2 x  1  cot 2 x  csc x   1  cot 2 x ; cosecant and sine are negative in IV;
2
1
3
3 10
10
10
 1
 sin x 


csc x   1      

csc x
10
9
3
10
 3
Now find the three remaining trigonometric functions of x.
1
tan x 
 3
cot x
1 3 10
10

cos x  cot x  sin x  cos x    
3
10
10
1
sec x 
 10
cos x
1
Ch 5 Notes F07 O’Brien
LHS Trig 8th ed
B.
Using Identities to Rewrite Functions and Expressions
Example 3 Use identities to rewrite cot x in terms of sin x. (#44)
cot x 
cos x
and from sin 2 x  cos 2 x  1 we know cos x   1  sin 2 x
sin x
therefore, cot x 
 1  sin 2 x
.
sin x
Example 4 Rewrite the expression sec θ  cot θ  sin θ in terms of sine and cosine and simplify. (#50)
sec θ  cot θ  sin θ 
1 cos θ
cos θ  sin θ

 sin θ 
1
cos θ sin θ
cos θ  sin θ
Example 5 Use identities to rewrite the expression sin2x + tan2x + cos2x in terms of sec x. (#63)
sin2x + cos2x = 1, so sin2x + tan2x + cos2x = 1 + tan2x which equals sec2x
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Section 5.2
I.
II.
Verifying Trigonometric Identities
Verifying Trigonometric Identities
A.
An identity is an equation that is true for all of its domain values.
B.
To verify an identity, we show that one side of the identity can be rewritten to look
exactly like the other side.
C.
Verifying identities is not the same as solving equations. Techniques used in solving
equations, such as adding the same term to both sides or multiplying both sides by the same
factor, are not valid when verifying identities.
Hints for Verifying Trigonometric Identities
A.
Know the fundamental identities and their equivalent forms inside out and upside down.
Example 1
sin2x + cos2x = 1 is equivalent to cos2x = 1 – sin2x
Example 2
tan x 
sin x
is equivalent to sin x  cos x  tan x
cos x
B.
Start working with the more complicated side of the identity and try to turn it into the
simpler side. Do not work on both sides of the identity simultaneously.
C.
Perform any indicated operations such as factoring, squaring binomials, distributing,
or adding fractions.
Example 3
2 sin 2 x  3 sin x  1 can be factored to 2 sin x  1sin x  1 (#17)
2
Ch 5 Notes F07 O’Brien
LHS Trig 8th ed
Example 4
D.
1
1

can be added by getting a common denominator
sin x cos x
1
1
1 cos x
1 sin x cos x  sin x






sin x cos x sin x cos x cos x sin x sin x  cos x
Sometimes it is helpful to express all trigonometric functions on one side of an identity in
terms of sine and cosine.
tan x
 sin x (#34)
Example 5
Verify
sec x
sin x
tan x cos x sin x cos x



 sin x
1
sec x
cos x 1
cos x
E.
Fractions with a sum in the numerator and a single term in the denominator can be rewritten
as the sum of two fractions.
Example 6
Verify
1  cos x
 csc x  cot x
sin x
1  cos x
1
cos x


 csc x  cot x
sin x
sin x sin x
Fractions with a difference in the numerator and a single term in the denominator can be
rewritten as the difference of two fractions.
F.
Sometimes it is helpful to rewrite one side of the identity in terms of a single trigonometric
function.
sin 2 x
 sec x  cos x (#42)
Example 7
Verify
cos x
sin 2 x 1  cos 2 x
1
cos 2 x



 sec x  cos x
cos x
cos x
cos x cos x
G.
Multiplying both the numerator and denominator of a fraction by the same factor (usually the
conjugate of the numerator or denominator) may yield a Pythagorean identity and bring you
closer to your goal.
Example 8
Verify
1
1  sin x
.

1  sin x
cos 2 x
1
1
1  sin x 1  sin x 1  sin x




1  sin x 1  sin x 1  sin x 1  sin 2 x cos 2 x
3
Ch 5 Notes F07 O’Brien
LHS Trig 8th ed
H.
As you selection substitutions, keep in mind the side you are not changing. It represents your
goal. Look for the identity or function which best links the two sides.



Verify tan 2 x 1  cot 2 x 
Example 9
1
1  sin 2 x
.

1 
1
1

2
2
tan 2 x 1  cot 2 x  tan 2 x1 

tan
x

1

sec
x



cos 2 x 1  sin 2 x
 tan 2 x 
I.
If you get really stuck, abandon the side you’re working on and start working on the other
side. Try to make the two sides “meet in the middle.”
sec x  tan x 2  1  sin x
Example 10
1  sin x
working on left side:
working on right side:________
sec x  tan x2 
1  sin x

1  sin x
sec 2 x  2 sec x tan x  tan 2 x 
1  sin x 1  sin x


1  sin x 1  sin x
1
cos 2 x
1
cos 2 x
2

1 sin x sin 2 x



cos x cos x cos 2 x
2 sin x
cos 2 x

1  2 sin x  sin 2 x
1  sin 2 x
sin 2 x
1  2 sin x  sin 2 x
cos 2 x
cos 2 x
1
cos 2 x

2 sin x
cos 2 x



sin 2 x
cos 2 x
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Section 5.3
I.
Sum and Difference Identities for Cosine
Cofunction Identities
cos (90° – θ) = sin θ
sin (90° – θ) = cos θ
cot (90° – θ) = tan θ
tan (90° – θ) = cot θ
csc (90° – θ) = sec θ
sec (90° – θ) = csc θ
Note: The angles θ and 90° – θ can be negative and / or obtuse.
Example 1
4
Ch 5 Notes F07 O’Brien
LHS Trig 8th ed
Example 2
Example 3
II.
Sum and Difference Identities for Cosine
cos (A + B) = cos A cos B – sin A sin B
[Functions stay together, operator changes.]
cos (A – B) = cos A cos B + sin A sin B
[Functions stay together, operator changes.]
Example 4
III.
Applying the Sum and Difference Identities
A.
Reducing cos (A – B) to a Function of a Single Variable
Example 5
B.
Finding cos (s + t) Given Information about s and t
Example 6
5
Ch 5 Notes F07 O’Brien
LHS Trig 8th ed
C.
Verification of an Identity
Example 7
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Section 5.4
I.
II.
Sum and Difference Identities for Sine
sin (A + B) = sin A cos B + cos A sin B
[Functions mix; sign stays.]
sin (A – B) = sin A cos B – cos A sin B
[Functions mix; sign stays.]
Sum and Difference Identities for Tangent
tan A  B 
III.
Sum and Difference Identities for Sine and Tangent
tanA  tanB
1  tanAtanB
tan A  B 
tanA  tanB
1  tanAtanB
Applying the Sum and Difference Identities
A.
Finding Exact Sine and Tangent Function Values
Example 1
Example 2
6
LHS Trig 8th ed
Ch 5 Notes F07 O’Brien
Example 3
B.
Writing Functions as Expressions Involving Functions of θ
Example 4
Example 5
C.
Finding Function Values and the Quadrant of A + B
Example 6
7
Ch 5 Notes F07 O’Brien
LHS Trig 8th ed
D.
Verifying an Identity Using Sum and Difference Identities
Example 7
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Section 5.5
I.
Double-Angle Identities
Double-Angle Identities
2 tan (A)
1  tan 2 (A)
sin (2A)  2 sin (A)  cos(A)
tan (2A) 
cos (2A)  cos 2 (A)  sin 2 (A)
cos (2A)  2 cos 2 (A)  1
A.
cos (2A)  1  2 sin 2 (A)
Finding Function Values of θ Given Information about 2θ
Example 1
8
LHS Trig 8th ed
B.
Ch 5 Notes F07 O’Brien
Finding Function Values of 2θ Given Information about θ
Example 2
C.
Using an Identity to Write an Expression as a Single Function Value or Number
Example 3
Example 4
D.
Verifying a Double-Angle Identity
Example 5
9
Ch 5 Notes F07 O’Brien
LHS Trig 8th ed
E.
Deriving a Multiple-Angle Identity
Example 5
II.
Product-to-Sum Identities
cos A  cos B 
1
cos A  B  cos A  B
2
sin A  sin B 
1
cos A  B  cos A  B
2
sin A  cos B 
1
sin A  B  sin A  B
2
cos A  sin B 
1
sin A  B  sin A  B
2
Using a Product-to-Sum Identity
Example 6
III.
Sum-to-Product Identities
 AB
 AB
sin A  sin B  2 sin 
  cos

 2 
 2 
 AB
 AB
sin A  sin B  2 cos
  sin 

 2 
 2 
 AB
 AB
cos A  cos B  2 cos
  cos

 2 
 2 
 AB
 AB
cos A  cos B  2 sin 
  sin 

 2 
 2 
Using a Sum-to-Product Identity
Example 7
10
Ch 5 Notes F07 O’Brien
LHS Trig 8th ed
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Section 5.6
I.
Half-Angle Identities
Half-Angle Identities
cos
A
1  cos A

2
2
sin
A
1  cos A

2
2
tan
A
sin A

2 1  cos A
tan
A 1  cos A

2
sin A
tan
A
1  cos A

2
1  cos A
In the first three half-angle identities, the sign is chosen based on the quadrant of
II.
A
.
2
Applying the Half-Angle Identities
A.
Using a Half-Angle Identity to Find an Exact Value
Example 1
B.
Finding Function Values of
θ
Given Information about θ
2
Example 2
11
LHS Trig 8th ed
C.
Ch 5 Notes F07 O’Brien
Finding Function Values of θ Given Information about 2θ
Example 3
D.
Using an Identity to Write an Expression as a Single Trigonometric Function
Example 4
Example 5
E.
Verifying an Identity
Example 6
12