Question

How do you find the exact values of the sine, cosine, and tangent of the angle `(5pi)/12`?

Answer 1

`"see explanation"`

Explanation:

`"using the "color(blue)"trigonometric identities"`

`•color(white)(x)sin(x+y)=sinxcosy+cosxsiny`

`•color(white)(x)cos(x+y)=cosxcosy-sinxsiny`

`"note that "(5pi)/12=pi/4+pi/6`

`rArrsin((5pi)/12)=sin(pi/4+pi/6)`

`rArrsin(pi/4+pi/6)`

`=sin(pi/4)cos(pi/6)+cos(pi/4)sin(pi/6)`

`=(1/sqrt2xxsqrt3/2)+(1/sqrt2xx1/2)`

`=sqrt3/(2sqrt2)+1/(2sqrt2)`

`=(sqrt3+1)/(2sqrt2)xxsqrt2/sqrt2=1/4(sqrt6+sqrt2)larrcolor(red)"exact value"`

`cos((5pi)/12)=cos(pi/4+pi/6)`

`rArrcos(pi/4+pi/6)`

`=cos(pi/4)cos(pi/6)-sin(pi/4)sin(pi/6)`

`=(1/sqrt2xxsqrt3/2)-(1/sqrt2xx1/2)`

`=(sqrt3-1)/(2sqrt2)xxsqrt2/sqrt2`

`=1/4(sqrt6-sqrt2)larrcolor(red)"exact value"`

`tan((5pi)/12)=sin((5pi)/12)/cos((5pi)/12)`

`color(white)(xxxxxxx)=(sqrt6+sqrt2)/(sqrt6-sqrt2)xx(sqrt6+sqrt2)/(sqrt6+sqrt2)`

`color(white)(xxxxxxx)=(6+2sqrt12+2)/4`

`color(white)(xxxxxxx)=(8+4sqrt3)/4`

`color(white)(xxxxxxx)=2+sqrt3larrcolor(red)"exact value"`