If x1 is a quadrant II angle x2 is a quadrant III angle sinx

If x1 is a quadrant II angle, x2 is a quadrant III angle, sin(x1) = 5 13 , and tan(x2) = 3 4 , find sin(x1 + x2), sin(x1 x2), cos(x1 + x2), and cos(x1 x2). (a) sin(x1 + x2) (b) sin(x1 x2) (c) cos(x1 + x2) (d) cos(x1 x2)

Solution

sin(x1) = 5/13 x1 in in IIquad

cosx1 = -12/13

tanx1 = - 5/12

tanx2 = 3/4 x2 is IIIquad

sinx2 = -3/5 ; cosx2 = -4/5

i) sin(x1+x2) = sinx1cosx2 +sinx2cosx1

=(5/13)(-4/5) + ((-3/5)(-12/13)

=(-20 +36)/65

= 16/65

ii) sin(x1 -x2) =

sinx1cosx2 - sinx2cosx1

=(5/13)(-4/5) - ((-3/5)(-12/13)

=(-20 -36)/65

= -56/65

iii) cos(x1 -x2) = cosx1cosx2 +sinx1sinx2

=(-12/13)*(-4/5) +( 5/13*(-3/5)

= ( 48 - 15)/65 = 33/65

iv) cos(x1 +x2) =

cosx1cosx2 - sinx1sinx2

=(-12/13)*(-4/5) - ( 5/13*(-3/5)

= ( 48 + 15)/65 = 63/65