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MTH 112
Elementary Functions
Chapter 6
Trigonometric Identities, Inverse Functions, and
Equations
Section 2
Identities: Cofunction, Double-Angle, & Half-Angle
Review Identities
Identities from Chapter 5
–
–
–
–
Reciprocal relationships
Tangent & cotangent in terms of sine and cosine
Cofunction relationships
Even/odd functions
Identities from Chapter 6, Section 1
– Pythagorean
– Sum & Difference
Cofunction Relationships


sin   x   cos x
2



tan   x   cot x
2



cos  x   sin x
2



cot   x   tan x
2



sec  x   csc x
2



csc  x   sec x
2

Established in Chapter 5 for acute angles only.
Using the sum & difference identities, they can be
established for any real number.
Additional Cofunction Identities


sin  x     cos x
2



sec x     csc x
2



cos x     sin x
2



csc x     sec x
2



tan  x     cot x
2



cot  x     tan x
2

These can be established two ways …
– Visually using left/right shifts on the graphs.
– Algebraically using the sum/difference identities.
Double Angle Identities
sin 2x
= sin(x+x)
= sin x cos x + cos x sin x
= 2 sin x cos x
Double Angle Identities
cos 2x
= cos(x+x)
= cos x cos x - sin x sin x
= cos2x – sin2x
Can you use these results to
determine the graphs of …
= cos2x – (1 – cos2x)
2x
2
y
=
sin
= 2 cos x – 1
and
2
2
= (1 – sin x) – sin x
2x
y
=
cos
= 1 – 2 sin2x
Double Angle Identities
tan 2x
= tan(x+x)
= (tan x + tan x) / (1 – tan x tan x)
= 2 tan x / (1 – tan2x)
Double Angle Identities
Summary …
• sin 2x = 2 sin x cos x
• cos 2x = cos2x – sin2x
= 2 cos2x – 1
= 1 – 2 sin2x
• tan 2x = 2 tan x / (1 – tan2x)
The Quadrant of 2
Given the quadrant of  what be the quadrant of
2?
 in quadrant 1  2 is in quadrant 1 or 2
 in quadrant 2  2 is in quadrant 3 or 4
 in quadrant 3  2 is in quadrant 1 or 2
 in quadrant 4  2 is in quadrant 3 or 4
Does it matter if 0 ≤  < 2 or  is some other coterminal angle?
The Quadrant of 2
Given the quadrant of  and one of the trig
values of , what will be the quadrant of 2?
– Find sin  and cos .
– Use double angle formulas to find sin 2 and cos 2.
– The signs of these values will determine the
quadrant of 2.
Half Angle Identities
Since cos 2x = 2 cos2x – 1 …
• cos2x = (1 + cos 2x)/2
Substituting x/2 in for x …
• cos2(x/2) = (1 + cos x)/2
The ± is
determined by the
quadrant
containing x/2.
Therefore, …
x
1  cos x
cos 
2
2
OR
x
1  cos x
cos  
2
2
Half Angle Identities
Since cos 2x = 1 - 2 sin2x …
• sin2x = (1 - cos 2x)/2
Substituting x/2 in for x …
• sin2(x/2) = (1 - cos x)/2
The ± is
determined by the
quadrant
containing x/2.
Therefore, …
x
1  cos x
sin 
2
2
OR
x
1  cos x
sin  
2
2
Half Angle Identities
Since tan x = sin x / cos x …
x
1  cos x
tan  
2
1  cos x
Multiplying the top and bottom of the fraction inside of
this radical by either 1 + cos x or 1 – cos x produces
two other forms for the tan(x/2) …
x
sin x
1  cos x
tan 

2 1  cos x
sin x
Note that these last
two forms do not
need the ± symbol.
Why not?
Half Angle Identities
Summary …
x
1  cos x
cos  
2
2
x
1  cos x
sin  
2
2
Remember, the
choice of the ±
depends on the
quadrant of x/2.
x
1  cos x
sin x
1  cos x
tan  


2
1  cos x 1  cos x
sin x
Simplifying Trigonometric
Expressions
The identities of this section adds to the types of
expressions that can be simplified.
Identities in this section include …
– Cofunction identities
– Double angle identities
– Half angle identities