explain why sin 30 degrees is 0.5000

Question

Give an explanation.

Answer 1

Lindsay,

 

Here's a simple geometric proof:.  Please draw the diagram:

 

Start with an equilateral triangle ABC with apex at A.  Each of its angles is 60 degrees.  Let each side measure x.

 

Take the perpendicular from A meeting BC at D.  By RHS, the two triangles  ABD and ACD

are congruent and therefore AD bisects angle A and the side BC.  So angle BAD is 30 degrees and side BD is x/2

BD/AB=(x/2)/x=1/2 is the definition of sine BAD =sine 60/2  =sine 30

 

QED

 

It is interesting to note that the sine of some angles can be expressed as fractions, some with radicals but most are transcendental....As Andrew M points out. sine can take on any value from -1 to +1.  It is only the sine of special angles that are not transcendental..30, 60 , 45 and 90 degrees are well known to you and generally you will memorize.  It was a big question hundreds of years ago which angles could be constructed with ruler and compass and this geometric question is highly related to the question of which sines are not transcendental. 

 

 

Answer 2

Draw a right triangle with an angle of 30 degrees, using a protractor.  Now you might now have a protractor, so just imagine for a second that you did.  (Bear with me).  No matter what size you make the triangle, if you measure the side opposite the 30 degree angle and the hypotenuse, and the hypotenuse, and divide opposite by hypotenuse (remember that that is the definition of sine) you will get about 0.5.  The more accurately you draw, the closer it will be, and if you draw and measure perfect (impossible in real life) you will get exactly 0.5.  Because this works for all right triangles, no matter the size, with an angle of 30 degrees, then, without naming the triangle, we can say sin30 = 0.5.  The same method was used originally to get the trig function values for any angle, and this was used to make tables.  Later, mathematicians figured out rules to determine values for some angles exactly, without measurement, and ways to get approximations (really good ones that your calculator uses) to get other angles. 

Answer 3

Think of a circle centered at the origin, having radius of 1.  Imagine the sine representing the y, or vertical value, of any point on the circle.  The point (0,1) would represent 90 degrees, and the sine, y, is 1 while the cosine, x, is 0.  Zero degrees on the circle would be the point (1,0) and the cosine, or x value is 1 while the sine, or y value, is 0.

 

Any point on the circle in area I of the coordinate plane x,y is going to have a sine between 0 and 1 with the value rising as you progress counterclockwise up the circle towards 90 degrees.