Question

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

Answer 1

#{: (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) :}#