Answer:
[tex]\cos (x-y)=\dfrac{1}{2}=0.5[/tex]
Step-by-step explanation:
Use formula:
[tex]\cos (x-y)=\cos x\cos y+\sin x\sin y[/tex]
Since
[tex]\sin x=-\dfrac{1}{2}[/tex]
and angle x is in the fourth quadrant, then cos x is greater than 0 and is equal to
[tex]\cos x=\sqrt{1-\sin ^2x}=\sqrt{1-\left(-\dfrac{1}{2}\right)^2}=\sqrt{1-\dfrac{1}{4}}=\sqrt{\dfrac{3}{4}}=\dfrac{\sqrt{3}}{2}[/tex]
Since
[tex]\cos y=\dfrac{\sqrt{3}}{2}[/tex]
and angle y is in the first quadrant, then sin y is greater than 0 and is equal to
[tex]\sin y=\sqrt{1-\cos ^2y}=\sqrt{1-\left(\dfrac{\sqrt{3}}{2}\right)^2}=\sqrt{1-\dfrac{3}{4}}=\sqrt{\dfrac{1}{4}}=\dfrac{1}{2}[/tex]
Hence,
[tex]\cos (x-y)=\cos x\cos y+\sin x\sin y=\dfrac{\sqrt{3}}{2}\cdot \dfrac{\sqrt{3}}{2}+\left(-\dfrac{1}{2}\right)\cdot \dfrac{1}{2}=\dfrac{3}{4}-\dfrac{1}{4}=\dfrac{1}{2}[/tex]
