Sines And Cosines Reference Sheet
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5Sines and Cosines for Fractions of π
The exact values of the trig ratios sin x and cos x for x = π, π/2, π/3, π/4 and π/6 are
taught in schools i.e.
x
π
π/2 π/3 π/4 π/6
√
3
1 √
1
sin x
0
1
2
2
2
√
1
1 √
3
cos x −1
0
2
2
2
π
π
But what about sin
and cos
? These trig ratios can be obtained by finding expressions
5
5
for sin 5x and cos 5x in terms of powers of sin x and cos x i.e.
3
5
cos 5x = 5 cos x − 20 cos
x + 16 cos
x
3
5
sin 5x = 5 sin x − 20 sin
x + 16 sin
x
. Setting sin 5x = 0 gives a fifth-order equation for sin x but when a factor of sin x is canceled,
2
π
2π
this gives a quadratic equation for sin
x and hence leads to the values for sin
and sin
. A
5
5
√
√
π
π
1
π
1
similar equation can be solved to get cos
. sin
=
10 − 2
5 and cos
=
6 + 2
5 =
5
5
4
5
4
√
1
1
2π
2π
+
5. The values of sin
and cos
can also be found from the analysis.
4
4
5
5
π
π
2
1 √
Exact values for sin
and cos
can be found by solving the equations 1 − 2 sin
x =
8
8
2
√
1 √
π
π
1
2
and 2 cos
x − 1 =
(both based on cos 2x = cos
. These give sin
=
2 −
2 and
4
8
2
2
√
π
1
π
π
cos
=
2 +
2. This is an example of finding the trig ratios for
if the values for
are
8
2
2n
n
known
2π
2π
Using this principle, the values of sin and cos of π/10 can be obtained from sin
and cos
5
5
as found earlier.
Values of sin and cos of π/12 can be obtained by two methods i.e. one involving the sin
and cos of π/3 and π/4 and the other involving the ratios at π/6. The first of these states that
sin π/12 = sin π/3 cos π/4 + sin π/3 cos π/4.
So, it would seem that there are two circumstances under which sin π/n and cos π/n can be
found.
1). If n = 2m i.e. if n is even and sin π/m and cos π/m are known then sin π/n and cos π/n
can be found e.g. sin and cos of π/10 are known so sin and cos of π/20 can be found.
2). If 1/n can be expressed as 1/a − 1/b where the trig ratios can be calculated for π/a and
π/b, then the ratios can be calculated for π/n. For example, as the trig ratios can be calculated
for π/4 and π/5 so they can be calculated for π/4 − π/5.
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