Addition of angles formulae:
cos(a + b) = cosacosb – sinasinb
sin(a + b) = cosasinb + sinacosb
Subtraction of angles formulae:
cos(a – b) = cosacosb + sinasinb
sin(a – b) = sinacosb - cosasinb
The double angle formulae:
cos(2a) = cos²a - sin²a
sin(2a) = 2cosasina
For the tangents of the new angles, divide sine by cosine.
Consider two angles a and b, with the following sine and cosine values.
sina = 1/3 and cosa = 2sqrt(2)/3
sinb = 2/3 and cosb = sqrt(5)/3
Find the following values as exact numbers.
sin(a + b)
cos(a + b)
tan(a + b)
sin(a - b)
cos(a - b)
tan(a - b)
sin(2a)
cos(2a)
tan(2a)
Answers in the comments.
sina = 1/3 and cosa = 2sqrt(2)/3
ReplyDeletesinb = 2/3 and cosb = sqrt(5)/3
Find the following values as exact numbers.
sin(a + b) = (sqrt(5) + 4sqrt(2))/9
cos(a + b) = (2 sqrt(10) - 2)/9
tan(a + b) = (sqrt(5) + sqrt(2))/2
sin(a - b) = (sqrt(5) - 4sqrt(2))/9
cos(a - b) = (2 sqrt(10) + 2)/9
tan(a - b) = (sqrt(2) - sqrt(5))/2
sin(2a) = 4sqrt(2)/9
cos(2a) = 7/9
tan(2a) = 4sqrt(2)/7
Mr. Hubbard can you give examples using Cos3alpha and Sin3aplpha using the sina= 1/3 and cosa=2sqrt(2/3)??
ReplyDeleteHi, Courtney! Sure thing.
ReplyDeletesin(3a) = sin (a + 2a), so use the addition formulas with the numbers
sin(a) = 1/3
cos(a) = 2sqrt(2)/3
sin(2a) = 4sqrt(2)/9
cos(2a) = 7/9
cos(3a) = 14sqrt(2)/27 - 4sqrt(2)/27 =
10sqrt(2)/27
sin(3a) = 7/27 + 16/27 =
23/27