Question

daca x apartine (0,pi/2 ),y apartine (pi/2 ,pi) si sin x=4/5 , cos y=-3/5 sa se calculeze sin (x-y),cos(x-y) ,cos(x+y)

Answer 1

sinx=4/5⇒cosx=3/5
cosy=-3/5⇒siny=4/5


sin(x-y)=sinxcosy-sinycosx=3/5·(-4/5)-4/5·3/5=-12/25-12/25=-24/25

cos(x-y)=cosxcosy+sinxsiny=3/5·(-3/5)+4/5·4/5=-9/25+16/25=7/25


cos(x+y)=cosxcosy-sinxsiny=-9/25-16/25=-25/25=-1

Answer 2

x€(0,π/2), sin x=4/5
sin²x+cos²x=1=>cos²x=1–sin²x=1–(4/5)²=1–16/25=25/25–16/25=9/25
cos x=±√(9/25)
x€(0,π/2) primul cadran
Deci cos x=√(9/25)=3/5

y€(π/2,π), cos y=–3/5
sin²y+cos²y=1=>sin²y=1–cos²y=1–(–3/5)²=1–9/25=25/25–9/25=16/25
sin y=±√(16/25)
y€(π/2,π) al doilea cadran
Deci sin y=√(16/25)=4/5

sin(x–y)=sin x*cos y–cos x*sin y= 4/5*(–3/5)–3/5*4/5=–12/25–12/25=–24/25

cos(x–y)=cos x*cos y+sin x*sin y= 3/5*(–3/5)+4/5*4/5=–9/25+16/25=7/25

cos(x+y)=cos x*cos y–sin x*sin y=3/5*(–3/5)–4/5*4/5=–9/25–16/25=–25/25=–1