SOLUTION: Why does sin 30° = 150° (or sin π/6 = sin 5π/6)? Refer to both the unit circle and the graph of the sine curve.

Algebra ->  Trigonometry-basics -> SOLUTION: Why does sin 30° = 150° (or sin π/6 = sin 5π/6)? Refer to both the unit circle and the graph of the sine curve.      Log On


   

Question 991180: Why does sin 30° = 150° (or sin π/6 = sin 5π/6)? Refer to both the unit
circle and the graph of the sine curve.
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
sine (30) = .5
sine (150) = .5
why are they the same.
it's because the reference angle for 150 is equal to 30.

that reference angle is the angle within the triangle formed from dropping a perpendicular to the x-axis of the unit circle.

here's a sketch of 30 degrees and 150 degrees on the unit circle and the triangles formed.

$$$

the 30 degree angle is in quadrant 1.

the 150 degree angle is in quadrant 2.

the internal angle of the triangle in quadrant 2 is equal to 180 - 150 which is equal to 30 degrees.

that is called the reference angle.

it is equivalent to the same angle in quadrant 1.

the sine of 30 degrees is equal to .5

the sine of 150 degrees is equal to .5

you can confirm using your calculator.

the graph of sin(x) and the intersection of that graph with sin(30) is shown below:

$$$

the angles of interest to you are 30 and 150.

you can see that the sine is .5 for both of them.

the other two angles are in quadrant 3 and quadrant 4.

the sine is the same except the sine is negative.

that's because sine is positive in quadrants 1 and 2 and is negative in quadrants 3 and 4.

on the unit circle, sine is equal to y / 1.

in quadrants 1 and 2, y is positive.

in quadrants 3 and 4, y is negative.

a 30 degree angle in quadrant 2 is 180 - 30 = 150.
a 30 degree angle in quadrant 3 is 180 + 30 = 210.
a 30 degree angle in quadrant 4 is 360 - 30 = 330.

if you are given the larger angles, you find the reference angle as follows:

in quadrant 2, the reference angle is 180 - 150 = 30.
in quadrant 3, the reference angle is 210 - 180 = 30.
in quadrant 4, the reference angle is 360 - 330 = 30.