How do you find the exact value of cos 7pi/4?

3 Answers

#cos(5.49778714377)=0.70710678117#.

Explanation:

Evaluate #7xxpi# then divide that by #4# first
So #7xxpi# is #7xxpi# or #21.9911485751#

#7xxpi=21.9911485751#

Now divide #7xxpi# by #4#

#21.9911485751/4=5.49778714377#

That means #cos (7)(pi)/4# is #cos(5.49778714377)#

#cos(5.49778714377)=0.70710678117#.

Apr 16, 2016

First, convert to degrees (for many people, these are more convenient to work with).

Explanation:

The conversion factor between radians and degrees is #180/pi#

#(7pi)/4 xx 180/pi#

#=315^@#

Now, this is a special angle, which can be found by using the special triangles.

But first, we must determine the reference angle of #315^@#. The reference angle #beta# of any positive angle #theta# is within the interval #0^@ <= beta < 90^@#, linking the terminal side of #theta# to the x axis. The closest intersection with the x axis for #315^@# would be at #360^@#: #360^@ - 315^@ = 45^@#. Our reference angle is #45^@#.

We now know that we must use the #45-45-90; 1, 1 sqrt(2)# triangle, as shown in the following graphic.

http://www.shmoop.com/trig-functions/special-trig-angle-obtuse.html

Now, it's just a matter of applying the definition of cos to find the wanted trig ratio.

#cos =# adjacent/hypotenuse

#cos = 1/sqrt(2)#, or #0.707#, as a fellow contributor stated. However, for the purpose of this problem, I think your teacher would be looking for an exact value answer: #cos((7pi)/4) = 1/sqrt(2)#

Hopefully this helps!

Apr 16, 2016

#sqrt2/2#

Explanation:

Trig unit circle and trig table -->
#cos ((7pi)/4) = cos (-pi/4 + (8pi)/4) = cos (-pi/4 + 2pi) = #
#cos (-pi/4) = cos (pi/4) = sqrt2/2#