Trig Distributive Problem - # 3

[Jason76]

$\displaystyle (\dfrac{\pi }{4})[[4 \sin \dfrac{\pi}{4}] + [4 \sin \dfrac{\pi }{2}] + [4 \sin \dfrac{3\pi }{4}] + [4 \sin \pi] + [4 \sin \dfrac{5\pi }{4}] + [4 \sin \dfrac{3\pi }{2}]]$ What does this come out to?

My attempt:

$\displaystyle (\dfrac{\pi }{4})[[4 (.707106781)] + [4 (1)] + [4(.70710678118)] + [4 (0)] + [4 (-.7071067811)] + [4(-1)]]$

 

[Deleted member 4993]

$\displaystyle (\dfrac{\pi }{4})[[4 \sin \dfrac{\pi}{4}] + [4 \sin \dfrac{\pi }{2}] + [4 \sin \dfrac{3\pi }{4}] + [4 \sin \pi] + [4 \sin \dfrac{5\pi }{4}] + [4 \sin \dfrac{3\pi }{2}]]$ What does this come out to?

My attempt:

$\displaystyle (\dfrac{\pi }{4})[[4 (.707106781)] + [4 (1)] + [4(.70710678118)] + [4 (0)] + [4 (-.7071067811)] + [4(-1)]]$

Now finish it.....

 

[HallsofIvy]

$\displaystyle (\dfrac{\pi }{4})[[4 \sin \dfrac{\pi}{4}] + [4 \sin \dfrac{\pi }{2}] + [4 \sin \dfrac{3\pi }{4}] + [4 \sin \pi] + [4 \sin \dfrac{5\pi }{4}] + [4 \sin \dfrac{3\pi }{2}]]$ What does this come out to?

My attempt:

$\displaystyle (\dfrac{\pi }{4})[[4 (.707106781)] + [4 (1)] + [4(.70710678118)] + [4 (0)] + [4 (-.7071067811)] + [4(-1)]]$

It might be better to write it as
$\displaystyle \pi\left[\dfrac{\sqrt{2}}{2}+ 1+ \dfrac{\sqrt{2}}{2}+ 0- \dfrac{\sqrt{2}}{2}- 1\right]$
where I have factored the "4" out of the sum, canceled it with the 4 in the denominator, and written the exact values for the sines.