Sec 6.2 Sum, Difference and Double Angle Identities -­‐  Page 1 of 3  

Math 12 Name _______________ Date ________________ Sec 6.2 Sum, Difference and Double Angle Identities 1. Sum and Difference Identities

In mathematics, we often combine the operations of addition or subtraction. These operations frequently give different results if they are carried out in different orders. For example:

1. Triple a sum 3(𝑎 + 𝑏) vs sum of the numbers tripled 3a + 3b

2. Square root of a sum 𝑎 + 𝑏 vs sum of the square roots 𝑎 + 𝑏

3. square of a number (𝑎 + 𝑏)! vs sum of the squares 𝑎! + 𝑏!

4. reciprocal of a sum 1a+ b

vs sum of the reciprocals 1a+1b

5. sine of a sum sin(𝛼 + 𝛽) vs sum of the sines sin(𝛼)+ sin  (𝛽) Only the first pair of expression are equal by applying the distributive property. In general, function operations are not distributive over addition Sum and Difference Identities sin(𝛼 + 𝛽) = sin(𝛼 − 𝛽) = cos(𝛼 + 𝛽) = cos(𝛼 − 𝛽) = tan(𝛼 + 𝛽) = tan(𝛼 − 𝛽) =

If the two angles are equal, then these identities reduce to two identities for sin 2𝛼, cos 2𝛼, and tan 2𝛼. Proofs 1) sin 2𝛼 = 2) cos 2𝛼 =

Sec 6.2 Sum, Difference and Double Angle Identities -­‐  Page 2 of 3  

3) cos 2𝛼 = 4) cos 2𝛼 = 5) tan 2𝛼 =

Double-Angle Identities

sin 2𝛼 = tan 2𝛼 cos2α = Examples 1 - Write each expression as a single trigonometric function. Evaluate, if possible.

a) sin6xcos x − cos6xsin x b) cos 7π12cosπ

3+ sin 7π

12sin π

3

c) 18sin3xcos3x d) 10sin π6cosπ

6

Sec 6.2 Sum, Difference and Double Angle Identities -­‐  Page 3 of 3  

e) 1− 2sin2 π6

f) 10cos2 π3− 5

Examples 2 - Expand. a) 7sin2θ b) cos(2π +θ ) 2. Determine Exact Trigonometric Values for Angles

Recall the special triangles: Useful angles:

Examples 3 - Determine the exact value for each expression.

a) cos 5π12

b) tan15°