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

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

Additional Cofunction Identities 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 = 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

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)

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?

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.

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

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

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?

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.

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