Can someone explain why sin(30°) = sin(150°)

Question

Can someone explain why sin(30°) = sin(150°) (or ). Refer to both the unit circle and the graph of the sine curve. Give as much detail as you can. 

Answer 1

From zero to 180 degrees, the sine graph goes from zero to a maximum 1 at 90 degrees, then back down to zero at 180 degrees. It's a symmetric hill, with height rising from minimum 0 to maximum 1, then 1 back to 0. Top of the hill is 1 at 90 degrees. 30 degrees is 60 less than 90. 150 is 60 greater than 90. 30 and 150 are opposite sides of the hill, at the same distance down from the top. Cos 30 = 1/2 Cos 150 is 1/2. Going from the origin to the top, you're half way up on the left side, then you're half way down on the right side.

On a unit circle, 30 degrees is the somewhat familiar 2:1:square root of 3 right triangle, with hypotenuse 1/2 of 2, adjacent side is 1/2 of the square root of 3 and the opposite side is 1/2 of 1 = 1/2

sine of 30 is opposite side over hypotenuse or 1/2 over 1 = 1/2

On the unit circle, 150 degrees gives the same right triangle, only in the 3rd quadrant, so that

sine 150 = opposite side over hypotenuse = 1/2 over 1 = 1/2. 150 degrees is 180-30 degrees.

The right triangle in the 3rd quadrant is 30 degrees when you drop a vertical down from the end of

the hypotenuse on the unit circle.

Another way to think of this is take the sine curve and shift it left by 90 degrees. That gives

the cosine curve which is an even function. In even functions. the graph is symmetric about the

y axis. f(x) = f(-x) cosx = cos(-x) What had been 30 degrees is now -60 degrees, with the

leftward 90 degree shift 30-90=-60 Cos(-60)=Cos(60) because it's an even function. Cos(60)

shifted back to the right by 90 degrees is sin(60+90) or sin 150 Shift right 90 degrees and

the cosine function becomes a sine function. Cos(-60) shifted to the right 90 degrees is

Sin(-60+90)=Sin30 Sin30=Sin150 just as Cos(-60)=Cos(60)