Question

How do you find the exact functional value tan 165 using the cosine sum or difference identity?

Answer 1

`sqrt(2-sqrt3)/-sqrt(2+sqrt3)`

Explanation:

`sin(165)=sin(180-15)=sin(15)=sqrt((1-cos2(15))/2)`=`sqrt((1-cos30)/2)=sqrt((1-sqrt3/2)/2)=sqrt(2-sqrt3)/2`
`cos(165)=cos(180-15)=-cos(15)=-sqrt((1+cos2(15))/2`=`-sqrt((1+cos30)/2)=-sqrt((1+sqrt3/2)/2)=-sqrt(2+sqrt3)/2`
`tan(165)=sin(165)/cos(165)=sqrt(2-sqrt3)/2/-sqrt(2+sqrt3)/2=sqrt(2-sqrt3)/-sqrt(2+sqrt3)`

Answer 2

Find tan (165)

Ans: `(1 - sqrt3)/(1 + sqrt3)`

Explanation:

Apply the trig identity: `tan (a + b) = (tan a + tan b)/(1 - tan a.tan b)`
tan (165) = tan (45 + 120).
Trig table gives --> tan 45 = 1; `tan 120 = -sqrt3`

`tan (165) = (1 - sqrt3)/(1 + sqrt3)`

Check by calculator: tan (165) = - 0.27
`(1 - sqrt3)/(1 + sqrt3) = - 0.73/2.73 = - 0.27`. OK