Find the exact value under the given conditions sin alpha 1

Find the exact value under the given conditions. sin alpha = 12/13, 0

Solution

sin alpha=12/13      here alpha is in first quadrant

cos beta=21/29        here beta is in first quadrant

sin alpha=12/13

here opposite=12   and hypotenuse=13

Therefore adjacent=sqrt(13²-12²)   =   5

cos alpha=adjacent/hypotenuse=5/13

cos beta=21/29

Here adjacent=21   and   hypotenuse=29

Therefore opposite=sqrt(29²-21²) = 20

sin beta=opposite/hypotenuse = 20/29

cos(alpha + beta) = cos alpha cos beta - sin alpha sin beta

                             = (5/13)(21/29)-(12/13)(20/29) = (105-240)/377

                                                                               = -135/377