Answer to Question #139914 in Trigonometry for Michael

Question

Determine the exact value of each trigonometric ratio

a) sin 75 degree
b) tan 5pie over 12
c) sin -pie over 12

Answer 1

a)sin(75^o) = sin(30^o + 45^o) = sin(30^o)*cos(45^o) + sin(45^o)*cos(30^o) = ($1/2$)*($\sqrt2/2$ ) + $(\sqrt2/2)*\sqrt3/2)$ = $(\sqrt2/2)*((1+\sqrt3)/2) = 0.9659$

b)tan(5*$\pi$/12) = tan(3*$\pi$/12 + 2*$\pi$/12) = tan($\pi$/4 + $\pi$/6) = (tan($\pi$/4) + tan($\pi$/6))/(1-tan($\pi$/4)*tan($\pi$/6)) = ($1 +$$\sqrt3/3$$)/(1-1*\sqrt3/3)$ = 3.732

c)sin(-$\pi$/12) = sin(2$\pi$/12 - 3$\pi$/12) = sin(2$\pi$/12)*cos(3$\pi$/12) - sin(3$\pi$/12)*cos(2$\pi$/12) =

sin($\pi$/6)*cos($\pi$/4) - sin($\pi$/4)*cos($\pi$/6) = $(1/2)*(\sqrt2/2) - (\sqrt2/2)*(\sqrt3/2) = (\sqrt2/2)*(1-\sqrt3) = -0.5176$