Question #75094

1 Answer
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