Question

Please help!!!! sinx= -1/2, and cosy= sqrt 3/2. if angle x is the fourth quadrant and angle y is in the first quadrant the value of cos(x-y) is??

Answer 1

`\cos (x-y)=(1)/(2)=0.5`

Explanation:

Use formula:

`\cos (x-y)=\cos x\cos y+\sin x\sin y`

Since

`\sin x=-(1)/(2)`

and angle x is in the fourth quadrant, then cos x is greater than 0 and is equal to

`\cos x=√(1-\sin ^2x)=\sqrt{1-\left(-(1)/(2)\right)^2}=\sqrt{1-(1)/(4)}=\sqrt{(3)/(4)}=(√(3))/(2)`

Since

`\cos y=(√(3))/(2)`

and angle y is in the first quadrant, then sin y is greater than 0 and is equal to

`\sin y=√(1-\cos ^2y)=\sqrt{1-\left((√(3))/(2)\right)^2}=\sqrt{1-(3)/(4)}=\sqrt{(1)/(4)}=(1)/(2)`

Hence,

`\cos (x-y)=\cos x\cos y+\sin x\sin y=(√(3))/(2)\cdot (√(3))/(2)+\left(-(1)/(2)\right)\cdot (1)/(2)=(3)/(4)-(1)/(4)=(1)/(2)`