Question

How do you find the exact value of the sine, cosine, and tangent of the number -10pie/3, without using a calculator?

Answer 1

Find sin, cos, and tan of `((-10pi)/3)`

Explanation:

The trig unit circle and Trig Table of Special Arcs give -->
`sin ((-10pi)/3) = sin ((-4pi)/3 - 2pi) = sin ((2pi)/3) = sqrt3/2`
`cos ((-10pi)/3) = cos ((2pi)/3) = - 1/2`
`tan ((-10pi)/3) = tan ((2pi)/3) = sin/(cos) = (sqrt3/2)(-2/1) = - sqrt3`

Answer 2

`sin(-(10pi)/3) =sqrt(3)/2 color(white)("XXXXX")cos(-(10pi)/3)=-1/2color(white)("XXXXX")tan(-(10pi)/3)=-sqrt(3)`

Explanation:

Moving clockwise around the origin we can see that
`-(10pi)/3` is equivalent to a reference angle of `pi/3` in Quadrant II

The values of `sin(-(10pi)/3)`, `cos(-(10pi)/3)` and `tan(-(10pi)/3)`
follow directly from this diagram and the definitions of these functions

Note: I assumed the "e" in "-10pie/3" was extraneous and not intended to be the natural logarithm base.