How do you find the exact functional value sin375 using the cosine sum or difference identity?

2 Answers
Oct 7, 2015

#sin(375)=sin(15)=(sqrt(3)-1)/(2sqrt(2))#

Explanation:

I'd use the identity #cos(b) sin(a)+cos(a) sin(b)#, with #a=360°#, and #b=15°#, since both are known values:

  • #cos(360)=cos(0)=1#
  • #sin(360)=sin(0)=0#
  • #cos(15)=(sqrt(3)+1)/(2sqrt(2))#
  • #sin(15)=(sqrt(3)-1)/(2sqrt(2))#

Plugging this value into the equality gives

#sin(375)=sin(360+15)=cos(15)*sin(360)+sin(15)*cos(360)=cos(15)*0+sin(15)*1#

So, we just discovered in a quite unpractical way that #sin(375)=sin(15)=(sqrt(3)-1)/(2sqrt(2))#: a very much easier way to notice this would have been using the periodicity of the sine function: #sin(x)=sin(360+x)#.

Oct 10, 2015

Find #sin 375#

Ans: #sqrt(2 - sqrt3)/2#

Explanation:

sin (375) = sin (15 + 360) = sin 15. Call sin (15) = sin a
#cos (2a) = cos (30) = sqrt3/2#
Use the trig identity: #cos 2a = 1 - 2sin^2 a#
#sqrt3/2 = 1 - 2sin^2 a#
#2sin^2 a = 1 - sqrt3 = (2 - sqrt3)/2#
#sin^2 a = (2 - sqrt3)/4#
#sin (15) = sin a = +- sqrt(2 - sqrt3)/2#
Since arc (15) is in Quadrant I, its sin is positive,
#sin (15) = sqrt(2 - sqrt3)/2#

Check by calculator.
sin 15 = 0.26
#sqrt(2 - sqrt3)/2 = 0.52/2 = 0.26#. OK