Basic trigonometry II
1. Sum and difference identities. The display below indicates a derivation of the sum identities for the
sine and cosine functions, provided that α, β and α + β are acute.
C
E
cos(α + β) = |OD| = |OA| − |DA| = |OA| − |CE|
= (cos α)|OB| − (sin α)|BC|
α
B
= cos α cos β − sin α sin β , and
1
sin(α + β) = |DC| = |AE| = |AB| + |BE|
= (sin α)|OB| + (cos α)|BC|
β
α
O
D
= sin α cos β + cos α sin β .
A
The derivation can be modified to cover the case when α + β is obtuse, and then extended to all real
values of α and β using the symmetries at the end of Basic trigonometry I. As doing so is tedious, it is
worth mentioning a general principle from which the identity for all real values of α and β follows from
the special case displayed above: If two rational functions of sines and cosines are equal on an interval of
positive length, then they are equal on their common domain. The general phenomenon underlying this fact
is called analytic continuation.
Replacing β by −β in the sum identities, and using that cos(−β) = cos β and sin(−β) = − sin β (see
the symmetry 6.1 in Basic trigonometry I), yields the difference identities for the sine and cosine functions,
which are given along with the sum identities below.
cos(α ± β) = cos α cos β ∓ sin α sin β
tan α ± tan β
1 ∓ tan α tan β
2. Double angle identities. Letting α = β = ϑ in the sum identities results in double angle identities,
which express a trigonometric function of 2ϑ in terms of trigonometric functions of ϑ.
2 tan ϑ
sin 2ϑ = 2 sin ϑ cos ϑ
cos 2ϑ = cos2 ϑ − sin2 ϑ
tan 2ϑ =
.
1 − tan2 ϑ
Using the (Pythagorean) identity cos2 ϑ + sin2 ϑ = 1 to eliminate sin2 ϑ, or cos2 ϑ, from the double angle
identity for the cosine, gives two new forms of this identity,
and
cos 2ϑ = 1 − 2 sin2 ϑ,
which are frequently useful.
3. Half angle identities. Solving the last two identities for cos2 ϑ, respectively sin2 ϑ, and then replacing
ϑ by 12 ϑ, gives the half angle identities,
cos2 12 ϑ =
1
(1
2
+ cos ϑ)
and
sin2 12 ϑ =
1
(1
2
cos(α + β) = cos α cos β − sin α sin β =
sin2
1
α
2
tan 12 ϑ =
sin ϑ
1 + cos ϑ
ϑ
B
12
13
=
33
,
65
=
1
(1
2
− cos α) =
1
2
1−
3
5
=
1
,
5
so
3
π
2
sin 21 α = ± 15
√
5.
1
1
4.2. Multiples of 12
π. Any integer multiple of 12
can be written the sum of an integer multiple of 13 and
5
2
1
1
an integer multiple of 4 . For example, 12 = 3 − 4 , so the difference identity for the sine function gives
√ √ √ 5
sin 12
π = sin 32 π − 14 π) = sin 23 π cos 14 π − cos 32 π sin 14 π = 21 3 21 2 − − 12 12 2
√
√
√ √
= 41 2 + 14 6, or 14 (1 + 3) 2.
p
π, where
q
1
5
q is a power of 2, or the product of 3 and a power of 2. For example,
= 2 (1 + cos 4 π) = 12 (1 −
p
√
√
1√
2) = 14 (2 − 2), and 21 π < 58 π < π where the cosine is negative, so cos 85 π = 21 2 − 2. To take
2
√ √
1
1
1
1
1
another example, writing 12 π as 3 π − 4 π, and using difference identities, gives cos 12 π = 4 (1+ 3) 2
√
1
1 √
and sin 12 π = 4 ( 3 − 1) 2. (Exercise: Make the calculations required to verify these results.) So the
half angle identity for the tangent gives, after rationalizing the denominator and simplifying the result,
√
1
1 √
sin 24
π
( 3 − 1) 2
√
√
√
4
1
π=
=
tan 24
√
√ = −2 + 2 − 3 + 6.
1
1 + cos 24
1 + 41 (1 + 3) 2
π
5. Converting products to sums. A products of sines and/or cosines can be expressed as a sum (or, as
a difference) by adding or subtracting difference and sum identities, and then solving for the product that
remains on the right hand side of the result. Exercise: Use this technique to derive the following identities.
cos α cos β = 12 cos(α + β) + cos(α − β) .
sin α sin β = 12 cos(α − β) − cos(α − β) .
sin α cos β = 12 sin(α + β) + sin(α − β)
6. Converting sums to products. A sum (or differences) of sines, or cosines, can be expressed as a product
by doubling one of the preceding identities, and then expressing the result in terms of the arguments of the
summands. Exercise: Use this technique to derive the following identities.
sin α ± sin β = 2 sin 21 (α ± β) cos 12 (α ∓ β)
cos α + cos β = 2 cos 12 (α + β) cos 21 (α − β)
cos α − cos β = −2 sin 21 (α + β) sin 21 (α − β)
7. Rational parametrization of a circle. The square of the half angle identity for the tangent can used to
express cos ϑ and sin ϑ as rational functions of z = tan 21 ϑ. Exercise: Show that
1
O
5
− 13
− − 54
Further determination requires more information. For example, if
< α < 2π then 34 π < 21 α < π,
√
where the sine function is positive, so sin 21 α = 51 5. On the other hand, if − 12 π < α < 0 then
√
− 14 π < 21 α < 0, where the sine function is negative, so sin 12 α = − 51 5..
− cos ϑ).
C
1ϑ
2
and the half angle identity for the sine function gives
Only the magnitude of cos 21 ϑ and sin 21 ϑ are determined by these identities, not whether they are positive or
negative. The tangent of 12 ϑ is completely determined by cos ϑ and sin ϑ. This can be seen by manipulating
the foregoing identities, or by reflecting on the figure below, and then appealing to analytic continuation.
A
3
5
cos2 85 π
Dividing each identity for the sine function by the corresponding identity for the cosine function, and then
expressing the right hand side of the result in terms of the tangent function (multiplying and dividing by the
reciprocal of cos α cos β), yields the sum and difference identities for the tangent function.
cos 2ϑ = 2 cos2 ϑ − 1
3
The sum identity for the cosine function gives,
4.3. Halving angles. The half angle identities can be used to evaluate trigonometric functions of
sin(α ± β) = sin α cos β ± cos α sin β
tan(α ± β) =
4.1. Elementary calculations. Suppose that P (3, −4) is on the terminal side of α and Q(−5, 12) is on the
terminal side of β (with both α and β in standard position). The distance from P to the origin is 5, and the
distance from Q to the origin is 13 (by Pythagoras’ formula). The double and half angle identities for the
tangent give
12
2 − 43
2 tan α
sin β
1
24
13
= 23 .
tan 2α =
,
and
tan
β
=
=
=
=
7
2
2
5
1 − tan2 α
1 + cos β
1 + − 13
1 − −4
D
The use of these identities for evaluating trigonometric functions is illustrated next.
2z
1 − z2
and
sin ϑ =
where
z = tan 12 ϑ.
1 + z2
1 + z2
This expresses points on the unit circle as rational functions of a variable z, and is of importance in the
integration of rational trigonometric functions. It also implies that the integer side lengths of a right angled
triangle are precisely those of the form 2pqr, r(p2 − q 2 ), r(p2 + q 2 ), where p, q and r are integers.
cos ϑ =