1. Section 2 Identities: Cofunction, Double-Angle, & Half-Angle
MTH 112 Elementary Functions Chapter 6 Trigonometric Identities, Inverse Functions, and Equations
Section 2
Identities: Cofunction, Double-Angle, & Half-Angle

2. 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

3. Cofunction Relationships
Established in Chapter 5 for acute angles only.
Using the sum & difference identities, they can be established for any real number.

4. Additional Cofunction Identities
These can be established two ways …
Visually using left/right shifts on the graphs.
Algebraically using the sum/difference identities.

5. Double Angle Identities
sin 2x
= sin(x+x)
= sin x cos x + cos x sin x
= 2 sin x cos x

6. Double Angle Identities
cos 2x
= cos(x+x)
= cos x cos x - sin x sin x
= cos2x – sin2x
= cos2x – (1 – cos2x)
= 2 cos2x – 1
= (1 – sin2x) – sin2x
= 1 – 2 sin2x
Can you use these results to determine the graphs of …
y = sin2x
and
y = cos2x

7. Double Angle Identities
tan 2x
= tan(x+x)
= (tan x + tan x) / (1 – tan x tan x)
= 2 tan x / (1 – tan2x)

8. 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)

9. Does it matter if 0 ≤  < 2 or  is some other coterminal angle?
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?

10. 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.

11. The ± is determined by the quadrant containing x/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
Therefore, …
The ± is determined by the quadrant containing x/2. OR

12. The ± is determined by the quadrant containing x/2.
Half Angle Identities
Since cos 2x = sin2x …
sin2x = (1 - cos 2x)/2
Substituting x/2 in for x …
sin2(x/2) = (1 - cos x)/2
Therefore, …
The ± is determined by the quadrant containing x/2. OR

13. Note that these last two forms do not need the ± symbol. Why not?
Half Angle Identities
Since tan x = sin x / 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) …
Note that these last two forms do not need the ± symbol. Why not?

14. Remember, the choice of the ± depends on the quadrant of x/2.
Half Angle Identities
Summary …
Remember, the choice of the ± depends on the quadrant of x/2.

15. 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