Question

How do you find the exact functional value cos 7pi/12 using the cosine sum or difference identity?

Answer 1

Find `cos ((7pi)/12)`

Ans: `(sqrt2 - sqrt6)/4`

Explanation:

`cos ((7pi)/12) = cos ((3pi)/12 + (4pi)/12) = cos (pi/4 + pi/3)`
Use trig identity: cos (a + b) = cos a.cos b - sin a.sin b

`cos a = cos (pi/4) = sqrt2/2 ; cos b = cos ((pi)/3) = 1/2`
`sin a = sin (pi/4) = sqrt2/2 ; sin a = sin (pi/3) = sqrt3/2`
`cos ((7pi)/12) = cos (pi/4 + pi/3)`=

`=(sqrt2/2)(1/2) - (sqrt2/2)(sqrt3/2) = (sqrt2 - sqrt6)/4`

Check by calculator:
(sqrt2 - sqrt6)/4 = - 0.259
cos ((7pi)/12) = cos 105 = - 0.259 OK