a  cos(a) sin(b) 
tan(a) cot(b) 
b  

radians  degrees  degrees  radians  
0  0  1  0  90 



15 
cos^{2}(15°) = [0; 1, 13, [1, 12]]

2 – 3
tan(15°) = [0; 3, [1, 2]]
tan^{2}(15°) = [0; 13, [1, 12]] 
75 


 18 
cos^{2}(18°) = [0; 1, 9, [2, 8]]

tan^{2}(18°) = [0; 9, [2,8]] 
72 


 22·5 
cos^{2}(22·5°) = [0; 1, 5, [1, 4]]

2 – 1 tan(22·5) = [0; [2]] tan^{2}(22·5) = [0; 5, [1,4]] 
67·5 
 
 30 
cos(30°) = [0; 1, 6, [2, 6]]
cos^{2}(30°) = [0; 1, 3] 
tan(30°) = [0; 1, [1,2]]
tan^{2}(30°) = [0; 3]  60 
 
 36 
cos(36°) = [0; 1, [4]]
cos^{2}(36°) = [0; 1, 1, 1, [8, 2]] 
tan^{2}(36°) = [0; 1, 1, [8, 2]]
 54 
 
 45 
cos(45°) = [0; 1, [2]]
cos^{2}(45°) = [0; 2] 
1  45 
 
 54 
cos^{2}(54°) = [0; 2, 1, [8, 2]]

tan^{2}(54°) = [1; 1, [8, 2]]
 36 
 
 60 
cos(60°) = [0; 2]
cos^{2}(60°) = [0; 4] 
3 tan(60°) = [1; [1,2]] tan^{2}(60°) = 3 
30 
 
 67·5 
cos^{2}(67·5) = [0; 6, [1, 4]]

1 + 2 tan(67·5) = [2; [2]]
tan^{2}(67·5) = [5; [1,4]] 
22·5 
 
 72 
cos(72°) = [0; 3, [4]]
cos^{2}(72°) = [0; 10, [2, 8] ] 
tan^{2}(72°) = [9; [2, 8]]
 18 
 
 75 
cos^{2}(75°) = [0; 14, [1, 12]]

2 + 3 tan(75°) = [3; [1, 2]] tan^{2}(75°) = [13; [1,12]] 
15 
 
 90  0  infinity  0  0 
Continued fraction [a; b,c,d,...] means a + 

Angle  sine cosine  Angle  

90 
 0  
60 
 30  
45 
 45  
30 
 60  
0 
 90 
cos(9)  = 
 
cos(18)  = 
 
cos(27)  = 
 
cos(36)  = 
 
cos(54)  = 
 
cos(63)  = 
 
cos(72)  = 
 
cos(81)  = 

_{} = 
 and = 



This triangle is just a square cut along a diagonal. If the sides are of length 1, the diagonal is length 2. This gives the sin, cos and tan of 45°.  Here is an equilateral triangle where all sides and all angles are equal (to 60°). If the sides are of length 2, then when we cut it in half as shown, the two triangles have 60°, 30° and 90° angles with a side of length 1 and a hypotenuse of length 2. The other side is therefore of length 3. So we can read off the sin cos and tan of both 30° and 60°. 
Phi = = 1.618033988.. =  1 + 5  = 1 +  1 
2  Phi 
The upper triangle with angles 72°, 72° and 36° and sides of lengths 1, Phi and Phi
shows the trig values for 18° and 72°.
The lower triangle with angles of 36°, 36° and 108° and sides of lengths 1, 1 and Phi shows the trig values of 36° and 54°. 
a^{2}  = b^{2} + c^{2} – 2 b c cos(A)  
For our triangle on the left, the known sides are b=2 and c=2 and the angle between them is A=30°. The length of the third side, the base a, is therefore:  
a^{2}  = 2^{2} + 2^{2} – 2 x 2 x 2 x cos(30°)  
= 8 – 4 3  
= 2 (4 – 2 3)  
But (3 – 1)^{2} = 3 + 1 – 2 3 = 4 – 2 3 and so  
a^{2}  = 2 (3 – 1)^{2}  
Taking the squareroot:  
a  = 2 (3 – 1) which we can also write as  
= 2 (3 – 1) / 2  
Using this expressions for a, we can expand the triangle by a factor of 2, to get rid of the denominator. Finally, we put in a line from the top of the triangle to the centre of the base a to make two rightangled triangles. This will halve the side a and cut the triangle into two and gets rid of the factor 2 also. We then arrive at the triangle on the right which shows the sines and cosines of 75° and 15°: 
sin(x) = a / h cos(x) = b / h tan(x) = a / b cot(x) = b / a 

If we know the value of a trig function on two angles A and B, we can determine the trig function values of their sum and difference using the following identities:
sin( A + B )  = sin(A)cos(B) + cos(A)sin(B) 
sin( A – B )  = sin(A)cos(B) – cos(A)sin(B) 
cos( A + B )  = cos(A)cos(B) – sin(A)sin(B) 
cos( A – B )  = cos(A)cos(B) + sin(A)sin(B) 
If the two angles are the same (i.e. A=B) we get the sines and cosines of double the angle. Rearranging those formulae gives the formula for the sin or cosine of half an angle:
sin( 2A ) = 2 sin(A) cos(A) cos( 2A ) = 1 – 2 sin^{2}(A) cos( 2A ) = cos^{2}(A) – sin^{2}(A) cos( 2A ) = 2 cos^{2}(A) – 1 

Each of those angles is measured from the top most point of the circle when a vertical line is turned through that angle.
Each line from the base point
meets the circle at a point whose a height is 1 (72°), 1+Phi (60°), 2+Phi (54°), 2+2 Phi (45°),
2+3 Phi (36°), 3+3 Phi (30°) or 3+4 Phi (18°).
Do look at his pages for more fascinating information on 120 3D solids, of which we will also explore the most symmetrical 5
on our next page.
Carl Friedrich GAUSS (177  1855) looked at a similar problem which answers this question.
He investigated if there
was a method of constructing a regular polygon of n sides using only a pair of compasses
(to draw circles) and a straightedge
(a ruler with no markings). We know we can construct a regular polygon for all of the
values of n=3, 4, 5, 6, 8 and 10.
The Trig Formula section above contains a formula for the cosine of half an angle in terms of the cosine of the (whole) angle:
cos 
 = 
 = 

However, this page is about sines and cosines which have simpler expressions, so we will not expand on this except to say that it shows how we can always find an exact expression for the sine (or cosine) of half of any angle for which we have an exact sine (or cosine) expression.
cos
4=
2
2
cos
8=
2 +
2
2
cos
16=
2 +
2 +
2
2
cos
32=
2 +
2 +
2 +
2
2
In the next 2000 years no one found an exact geometric method for 7gons or 9gons but also no one had proved it was impossible to construct such regular polygons.
Then Gauss completely solved the problem while he was a student at Göttingen between 1795 and 1798. Gauss found the conditions on the primes and their powers to solve two equivalent problems:
For instance:
cos  8  = 
 
17  16 
where S = 
 and T = 
