We have to find the values of:
cosπ12;cos5π12;sinπ12 & sin5π12
We can write π12 as: π12=π3−π4
⇒sinπ12=sin(π3−π4)
Using the compound angle formula for sin, we get:
sin(π3−π4)=sinπ3cosπ4−cosπ3sinπ4
⇒sin(π3−π4)=√32⋅1√2−12⋅1√2
⇒sin(π3−π4)=√3−12√2
Similarly,
cosπ12=cos(π3−π4)
⇒cos(π3−π4)=cosπ3cosπ4+sinπ3sinπ4
⇒cos(π3−π4)=12⋅1√2+√32⋅1√2
⇒cos(π3−π4)=√3+12√2
Also, 5π12=π2−π12
⇒5π12 & π12 are complimentary angles.
⇒sin5π12=cosπ12 & cos5π12=sinπ12
⇒sin5π12=cosπ12=√3+12√2 & cos5π12=sinπ12√3+12√2=√3−12√2