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

2 Answers
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

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})