Question

Find exact value of sine,cosine,and tagent of 19*3.14/6?

Answer 1

`cos((19 \pi}/6 ) = cos({7 pi}/6) = -cos(\pi/6) = -\sqrt{3}/2`

`sin((19 \pi}/6 ) = sin({7 pi}/6) = -sin(\pi/6) = -1/2`

`tan((19 \pi}/6 ) = {-1/2}/{-\sqrt{3}/2} = 1/sqrt{3}`

Explanation:

The exact value of `cos ( { 19 times 3.14 }/ 6 )` won't have a closed form and will be an infinite non-repeating decimal.

The exact value of `cos((19 \pi}/6 )` gets us to the biggest cliche in trig, the 30/60/90 triangle.

#cos((19 \pi}/6 ) = cos((19 \pi}/6 - 2pi) = cos({7 pi}/6) = -cos({7 \pi}/6 - \pi) = -cos(\pi/6) = -\sqrt{3}/2 #

`pi/6` is of course `30^circ`

#sin((19 \pi}/6 ) = sin((19 \pi}/6 - 2pi) = sin({7 pi}/6) = -sin({7 \pi}/6 - \pi) = -sin(\pi/6) = -1/2 #

`tan((19 \pi}/6 ) = sin((19 \pi}/6 )/{ cos((19 \pi}/6) } = {-1/2}/{-\sqrt{3}/2} = 1/sqrt{3}`