Give that alpha lies in quadrant II and sin alpha 45 beta l

Give that alpha lies in quadrant II and sin alpha = 4/5, beta lies in quadrant III and tan beta = 5/12, find the exact value of sin (alpha + beta) tan (alpha - beta) cos 2 alpha

Solution

sin alpha=4/5      alpha is in second quadrant

opposite=4    hypotenuse=5   

adjacent=sqrt(hypotenuse²-adjacent²) = 3

cos alpha= adjacent/hypotenuse-3/5

tan alpha= -4/3

tan beta=5/12

opposite=5

adjacent=12

hypotenuse=sqrt(opposite²+ adjacent²) = 13

sin beta= -5/13,cos beta=-12/13

a. sin(alpha + beta)=sin alpha cos beta + cos alpha sin beta = (4/5)(-12/13)+(-3/5)(-5/13) = -33/ 65

b. tan(alpha-beta)= (tan alpha-tan beta)/(1 +tan alpha tan beta)= ((-4/3)-(5/12))/(1+(-4/3)(5/12))=-63/16

c. cos 2 alpha= 2cos²alpha -1=2(-3/5)²-1= -7/25