How do you find exact value of sin (19pi/12)?

2 Answers
Mr. Hsia
May 11, 2018

#(-sqrt(2) - sqrt(6))/4#

Explanation:

#sin((19pi)/12) = sin((15pi)/12 + (4pi)/12)#
# = sin((5pi)/4 + pi/3)#
# = sin((5pi)/4)cos(pi/3) + cos((5pi)/4)sin(pi/3)#
# = (-sqrt(2)/2)(1/2) + (-sqrt(2)/2)(sqrt(3)/2)#
# = (-sqrt(2)/4) + (-sqrt(6)/4)#
# = (-sqrt(2) - sqrt(6))/4#

Dean R.
May 11, 2018

# sin 285^circ= -1/4 (sqrt {2} + sqrt{6}) #

Explanation:

I like to complain that every trig problem uses one of two triangles, 30/60/90 or 45/45/90. This one uses both!

The other answer is fine. Let's do it in degrees here.

# {19 pi }/ 12 times 360^circ/{2pi } = 285^circ #

# sin 285^circ = sin( -360^circ + 285^circ ) = sin( -75^circ) = - sin 75^circ = -sin(30^circ + 45^circ)#

There they are.

# sin 285^circ= -sin(30^circ + 45^circ) #

#= -( sin 30^circ cos 45^circ + cos 30^circ sin 45^circ ) #

#= -((1/2) (sqrt{2}/2) + (sqrt{3}/2)(sqrt{2}/2)) #

#= -1/4 (sqrt {2} + sqrt{6}) #

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