HW 4.5.1.a: Sum and Difference Formulas Use the sum and difference formulas to find an exact value for each of the following. 1.sin 105°( ) 2.sin 135°( ) 3. cos(195°) 4.cos(315°)

5.sin 7π

12⎛⎝⎜

⎞⎠⎟

6.sin π

12⎛⎝⎜

⎞⎠⎟

7.cos 5π

12⎛⎝⎜

⎞⎠⎟

8. cos 11π

12⎛⎝⎜

⎞⎠⎟

Rewrite in terms of ( )sin x and ( )cos x .

9.cos x + 11π6

⎛⎝⎜

⎞⎠⎟

10.cos x − π4

⎛⎝⎜

⎞⎠⎟

11. sin x − 7π6

⎛⎝⎜

⎞⎠⎟

12. sin x + 4π3

⎛⎝⎜

⎞⎠⎟

Solve each equation for all solutions.

13. sin 3x( )cos 6x( )− cos 3x( )sin 6x( ) =  22

14. sin 11x( )cos 6x( )− cos 11x( )sin 6x( ) =  32

15. ( ) ( ) ( ) ( )cos 2 cos sin 2 sin 1x x x x+ = 16. ( ) ( ) ( ) ( ) 3cos 5 cos 3 sin 5 sin 32

x x x x− =

17. How could you evaluate tan 13π12

⎛⎝⎜

⎞⎠⎟ if you did not know the sum and difference formula for

tangent? 18. Prove the sine and cosine cofunction identities using the sum and difference formulas.

Selected Answers: 1. sin(105°) = sin(60°+45°) = sin(60°)cos(45°)+cos(60°)sin(45°) = !

!∙ !!+ !

!∙ !!= !! !

!

3. cos(195°) = cos(150° + 45°) = cos(150°)cos(45°)-sin(150°)sin(45°) = !( !! !)!

5. sin !!!"

= sin !!+ !

!= sin !

!cos !

!+ cos !

!sin !

!= !! !

!

7. cos !!!"

= cos !!+ !

!= cos !

!cos !

!− sin !

!sin !

!= !! !

!

9. cos 𝑥 + !!!!

= cos 𝑥 cos !!!!

− sin 𝑥 sin !!!!

= !!cos 𝑥 + !

!sin 𝑥

11. sin 𝑥 − !!!

= sin 𝑥 cos !!!

− cos 𝑥 sin !!!

= − !!sin 𝑥 + !

!cos 𝑥

13. sin 3𝑥 cos 6𝑥 − cos 3𝑥 sin 6𝑥 = !!

sin 3𝑥 − 6𝑥 = !!

sin −3𝑥 = !!

−sin 3𝑥 = !!

, sin 3𝑥 = − !!

3𝑥 = !!!+ 2𝜋𝑘 or 3𝑥 = !!

!+ 2𝜋𝑘, where k is an integer

𝑥 = !!!"+ !!

!𝑘 or 𝑥 = !!

!"+ !!

!𝑘, where k is an integer

15. cos 2𝑥 cos 𝑥 + sin 2𝑥 sin 𝑥 = 1

cos 2𝑥 − 𝑥 = 1

𝑥 = 0+ 2𝜋𝑘, where k is an integer