Question

What is the exact value of sin330°?

Answer 1

@Frostbite

Answer 2

@Miracrown

Answer 3

For angles of any magnitude, first find out which quadrant your angle lies.

330 is between 270 and 360, thus quadrant 4.
The trig quadrant rules ASTC means 4th quadrant is positive only for the cosine ratio. This means sin of angles in quadrant 4 must then be negative. Thus your answer will be negative. But how do we find the value?

Well remember that angles in any quadrant can be reduced to a simpler angle that we are familiar with in quadrant 1.
Rewrite the angle as a difference of 360 for quadrant 4 as follows

sin(330) = sin (360 - ?) The ? is the angle you must subtract from 360 to give you 330, which is simply 30 degrees.

Thus we can say sin(330) = sin(360-30)
Almost done, we can now forget the 360 and just write it as sin(30) BUT with the quadrant sign attached. For Quadrant 4 we said sine was negative so
sin(330) = - sin(30)

you should see we changed the large angle to an acute angle we like.
sin(30) is 1/2 as an exact ratio thus sin(330) = - sin(30) = -1/2

For other quadrants you will need to either find the number to add to 180 (Quad 3), subtract from 180 (Quad 4).

So sin 225 for example is a quad 3 angle, and by ASTC rules is negative.
So we write sin 225 = sin (180+?) where ? = 45
We can remove the 180 and write sin 225 = - sin 45 once you insert the negative sign and then apply the exact value as -1/sqrt(2)

sin 120 same thing, quadrant 2 angle and positive according to ASTC.
We write sin(120) = sin(180 - ?) with ? being 60 so sin(120) = sin (180-60) = sin60 removing the 180 and taking into account the positive sign.