Question #4bd79

1 Answer
Bill K.
Sep 27, 2017

Yes. It's probably looking for the answers in exact form. The answers are #x((2pi)/3)=4sqrt(3)#, #v((2pi)/3)=-4#, and #a((2pi)/3)=-4sqrt(3)#.

Explanation:

The reasons are that #sin((2pi)/3)=sqrt(3)/2# and #cos((2pi)/3)=-1/2#.

These facts can be derived from drawing a 30-60-90 (degrees) triangle (with 30 degree angle at the top) whose hypotenuse has length 1 and whose base has length 1/2 (because its half the length of an equilateral triangle whose sides all have length 1). (See the figure below)

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The Pythagorean Theorem then implies that the height of the triangle is #sqrt(1^2-(1/2)^2)=sqrt(3/4)=sqrt(3)/2#. Now use SOH, CAH, TOA to say that #cos(60^{circ})=1/2# and #sin(60^{circ})=sqrt(3)/2#.

Since #(2pi)/3=120^{circ}# and #pi/3=60^{circ}# are supplementary angles, their cosines are opposites (#cos(120^{circ})=-cos(60^{circ})#).

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