Given sin theta = 2/3 and pi/2<theta<pi how do you find the value of other 5 other trigonometric functions?

1 Answer
Alan P.
Oct 20, 2015

#sin(theta) = 2/3color(white)("XXXXXXX")csc(theta)=3/2#
#cos(theta) = -sqrt(5)/3color(white)("XXXX")sec(theta)=-3/sqrt(5)#
#tan(theta)=-2/sqrt(5)color(white)("XXXX")cot(theta)=-sqrt(5)/2#

Explanation:

#pi/2 < theta < pi#
tells us that #theta# is in Quadrant II

#sin(theta)=2/3#
tells us that the side ratios are
#color(white)("XXX")#opposite side: #2#
#color(white)("XXX")#hypotenuse: #3#
and using the Pythagorean Theorem and the fact that we are in Q II
#color(white)("XXX")#adjacent side: #-sqrt(5)#

By definition:
#sin = ("opposite")/("hypotenuse")color(white)("XXXX")csc=("hypotenuse")/("opposite")#

#cos = ("adjacent")/("hypotenuse")color(white)("XXXX")sec=("hypotenuse")/("adjacent")#

#tan = ("opposite")/("adjacent")color(white)("XXXX")cot=("adjacent")/("opposite")#

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