If x a third quadrant angle and y an acute angle such that cos x=-(3/5) and sec y= (3/2) ,find the exact value of sin(x-y) , exact value of tan(x-y) and quadrant containing (x-y)?

1 Answer
Aug 20, 2015

Find sin (x - y), knowing cos x = -3/5 and sec y = 3/2

Explanation:

Apply the trig identity:
sin (x - y) = sin x.cos y - sin y.cos x
cosx=35 --> sin2x==1925=1625 --> sinx=45
cosy=23 -->sin2y=149 --> sin2y=59--> siny=53

sin(xy)=(45)(23)(53)(35)=815+3515=35815
Apply trig identity:
cos(xy)=cosx.cosy+sinx.siny=
=(35)(23)+(45)53)=6154515=
=45+615
tan(xy)=sincos=35845+6
The arc (x - y) is located in Quadrant III since its cos and its sin are both negative.
Check by calculator:
cosx=35 --> x = 360 - 126.86 = 233.14 deg (Quadrant III) ; cosy=23 --> y = 48.19 deg
Arc (x - y) = 233.14 - 48.19 = 184.95 deg --> sin (x - y) = -0.09
35815=0.09 OK