hw 4.5.1.a sum and difference formulas
TRANSCRIPT
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