Give the exact value of each expression in simplest radical

Give the exact value of each expression in simplest radical form. (a) cos 255 degree (b) tan (-7 pi/12)

Solution

1) cos225

cos(225)=cos(45+180)

Use the formula : cos(A +B) = cosAcosB -sinAsinB

=cos45cos180+sin45sin180
=sqrt(2)/2*(-1)+sqrt(2)*(0)
=-sqrt(2)/2+0
=-sqrt(2)/2

2) tan(-7pi/12) = -tan7pi/12 as tan(-x) = -tanx

Use the formula :

tan(x+y)=(tanx+tany)/(1-tanxtany)
let x=3pi/12=pi/4
let y=4pi/12=pi/3
tan(7pi/12) is in Quadrant II, therefore, it is negative

tanx=tan pi/4 =1
tany=tan pi/3 =sqrt(3)
tan(7pi/12) =tan(pi/4+pi/3) =(1+sqrt(3))/(1-sqrt(3))
= (1+sqrt(3)(1+sqrt(3)/(1+sqrt(3)(1-sqrt(3)
= (1+2sqrt(3)+3)/1-3
= (4+2sqrt(3))/-2
= -(2+sqrt(3))

tan(-7pi/12) = (2+sqrt(3))