# How do you find sine, cosine, tangent of 90^@ or 180^@ using the unit circle?

Mar 10, 2018

{: (sin(90^circ)=1,color(white)("xx"),sin(180^circ)=0), (cos(90^circ)=0,,cos(180^circ)=-1), (tan(90^circ)" is undefined",,tan(180^circ)=0) :}

#### Explanation:

For the unit circle we have the situations pictured below; ${90}^{\circ}$ and ${180}^{\circ}$ are limits as the hypotenuse approaches the (positive) vertical axis and the (negative) horizontal axis respectively. By definition:
color(white)("XXX")sin=(color(green)("opposite"))/(color(red)("hypotenuse"))

color(white)("XXX")cos=(color(blue)("adjacent"))/(color(red)("hypotenuse"))

color(white)("XXX")tan=(color(green)("opposite"))/(color(blue)("adjacent"))

As limits we can see that
color(white)("XXX"){: (sin(90^circ)=(color(green)1)/(color(red)1)=1,color(white)("xx"),sin(180^circ)=(color(green)0)/(color(red)(-1))=0), (,,), (cos(90^circ)=(color(blue)0)/(color(red)1)=0,,cos(180^circ)=(color(blue)(-1))/(color(red)1)=-1), (,,), (tan(90^circ)=(color(green)1)/(color(blue)0)" : undefined",,tan(180^circ)=(color(green)0)/(color(blue)(-1))=0) :}