Question #75094

1 Answer
mason m · Anjali G
Apr 16, 2017

#csc(2arctan(3/4))=25/24#

Explanation:

Because #sin# and #csc# are reciprocals:

#csc(2arctan(3/4))=1/sin(2arctan(3/4))#

Using the double angle identity #sin(2theta)=2sin(theta)cos(theta)#:

#=1/(2color(blue)(sin(arctan(3/4)))color(red)(cos(arctan(3/4)))#

We can find the values of #sin(arctan(3/4))# and #cos(arctan(3/4))# using a similar method.

Note that when #theta=arctan(3/4)#, then #tan(theta)=3/4#. That is, where #theta# is an angle in a right triangle, the side opposite #theta# is #3# and the leg adjacent to #theta# is #4#. The Pythagorean theorem tells us that the hypotenuse is #5#.

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We then see that:

#color(blue)(sin(arctan(3/4)))=sin(theta)="opposite"/"hypotenuse"=3/5#

#color(red)(cos(arctan(3/4)))=cos(theta)="adjacent"/"hypotenuse"=4/5#

So the original expression is:

#=1/(2color(blue)((3/5))color(red)((4/5)))#

#=25/24#

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