if Tan α= 21/20 and α lies in quadrant III, cos β = -5/13 and β lies in quadrant II, what is sin ( α + β ) ?

Question

use the given information to find the exact value of the expression if Tan α= 21/20 and α lies in quadrant III, cos β = -5/13 and β lies in quadrant II, what is sin ( α + β ) ?

Answer 1

Angle A in quadrant3:

y=opposite = 21 and x=adjacent=20

then hypotenuse = sqrt( 21^2 + 20^2)

= sqrt( 441 + 400)

= sqrt( 881)

So then sin A = 21/sqrt(881) = 21 * sqrt(881)/881

and cos A = 20/sqrt(881) = 20*sqrt(881) / 881

Angle B in Quadrant 2:

adjacent = -5 and hypotenuse = 13

then by pythagorean, y=opposie = 12

so then sin B = 12/13, cos B = -5/13 as given,

and tan B = -13/12

sin (A+B) = sin A cos B + cos A sin B

= 21 * sqrt(881)/881 * (-5/13) + 20*sqrt(881)/881 * (12/13)

= (-105/(13*881)) * sqrt(881) + (20/ (13*881))*sqrt(881)

= -85/(13*881) * sqrt(881)