13: Trigo Half Ratios

Picture

This is an interesting method to find the Sin, Cos, Tan of half angles i.e. if you know the measure of 30 degrees, this method will help you find the ratios for 15 degrees. The theory is as follows:

We draw an arc with centre at O and radius 1 unit. This arc cuts the x axis at (1,0) and as we turn this line by 30 degrees, we get the other Point  P. Thus we have an isosceles triangle here and the perpendicular dropped from the vertex, Origin here, bisects the opposite side.

So, we find the midpoint of the base of the triangle, Q, as shown in the diagram and since it is an isosceles triangle, the angle at the Vertex of the triangle is also bisected and thus we end up with 15 degrees. 

So, Q being midpoint of the base of the triangle we can find its coordinates easily. At the same time we know that the line OQ will have a slope equal to tan 15 and that is how we find its value as shown in the diagram. 


what if we change the angle to 45 and find tan 22.5 degree
By using the same method as above, we can find trigo ratios for any angle provided we know the values of the double of that angle. 

Here we show the general theorem that is developed for obtaining half angle ratios. 

Exercise
Since we know tan 15 now, find tan 7.5, also find tan 11.25