Question

Find the exact value by using a half-angle identity.

cos(5pi/12)

Answer 1

os 5π/12
cos ((5 pi/6)/2) = +/- sqrt( (1+cos (5 pi/6))/2 )
cos(5 pi/6) = -sqrt(3)/2
cos (5 pi/12) = +/- sqrt( (2/2-sqrt(3)/2)/2 )
cos (5 pi/12) = +/- sqrt( ((2-sqrt(3))/2)/2 )
cos (5 pi/12) = sqrt( ( (2-sqrt(3))/4 ) )

5 pi/12 is in quadrant 1 so we choose the + sign because cos is positive there

Answer 2

\(\bf \cfrac{5}{12}\to \cfrac{\cancel{ 3 }}{\cancel{ 12 }}+\cfrac{\cancel{ 2 }}{\cancel{ 12 }}\to \cfrac{1}{4}+\cfrac{1}{6}\qquad thus
\\ \quad \\
\cfrac{5\pi}{12}\to \cfrac{1\pi}{4}+\cfrac{1\pi}{6}\to \cfrac{\pi}{4}+\cfrac{\pi}{6}\qquad thus
\\ \quad \\
cos\left(\frac{5\pi}{12}\right)\to cos\left(\frac{\pi}{4}+\frac{\pi}{6}\right)\)

Answer 3

hhh ohh the half angle... shoot... so... check above then

Answer 4

ok, thank you!!