Download 1. Trigonometric Identities The Pythagorean Theorem, sin2 x + cos2

1. Trigonometric Identities
2
The Pythagorean Theorem, sin x + cos2 x = 1, has other forms, like
(1)
sin2 x cos2 x
1
+
=
⇒ tan2 x + 1 = sec2 x
2
2
cos x cos x
cos2 x
and
(2)
sin2 x cos2 x
1
+
=
⇒ 1 + cot2 x = csc2 x.
2
2
2
sin x sin x
sin x
Angle Addition:
(3)
sin(α + β) = sin α cos β + cos α sin β
(4)
cos(α + β) = cos α cos β − sin α sin β
The angle addition formulae give us the double angle formulae:
(5)
sin(2θ) = 2 sin θ cos θ
(6)
cos(2θ) = cos2 θ − sin2 θ
Because cos(2θ) = cos2 θ − (1 − cos2 θ) = 2 cos2 θ − 1, we have the half-angle formula
cos2 θ =
(7)
1 + cos(2θ)
.
2
Similarly, because cos(2θ) = 1 − sin2 θ − sin2 θ, we find the other half-angle formula
sin2 θ =
(8)
1 − cos(2θ)
.
2
2. Derivatives of Trigonometric Functions
We know the derivatives
d
(9)
sin x = cos x
dx
which allow us to calculate
and
d
cos x = − sin x,
dx
(10)
d
cos x
sin x
cos2 x + sin2 x
tan x =
+ sin x 2 =
= sec2 x
dx
cos x
cos x
cos2 x
(11)
d
− cos x
csc x =
= − csc x cot x
dx
sin2 x
(12)
d
sin x
sec x =
= sec x tan x
dx
cos2 x
(13)
− sin2 x − cos2 x
d
− sin x
− cos x
=
= − csc2 x
cot x =
+ cos x
dx
sin x
sin2 x
sin2 x
1
2
To compute
R
3. Useful Integrals
cos2 xdx,
(14)
Similarly, to compute
R
(15)
To compute
R
To compute
R
sin2 xdx, we use the half-angle formula for sines:
Z
Z
1 − cos(2x)
x sin(2x)
2
sin xdx =
dx = −
+C
2
2
4
dx
,
1−x2
Z
(16)
we use the half-angle formula for cosines:
Z
Z
1 + cos(2x)
x sin(2x)
cos2 xdx =
dx = +
+C
2
2
4
we use the method of partial fractions:
¶
Z µ
dx
1
1
1
1
=
+
dx = [ln(1 + x) − ln(1 − x)] + C
2
1−x
2
1+x 1−x
2
4. Integrals using Trigonometric Substitutions
dx
,
1+x2
we substitute
x = tan θ ⇒ dx = sec2 θdθ
(17)
to find
Z
(18)
To compute
R
√ dx ,
1−x2
dx
=
1 + x2
(20)
sec2 θ
dθ = θ + C = arctan x + C
sec2 θ
we substitute
(19)
to find
Z
x = sin θ ⇒ dx = cos θdθ
Z
dx
√
=
1 − x2
Z
cos θ
p
1 − sin2 θ
dθ = θ + C = arcsin x + C
5. Integration by Parts
The Product Rule states that d(uv) = udv + vdu. Integrating, we find
Z
Z
Z
Z
Z
Z
Z
(21)
d(uv) = udv + vdu ⇒ uv = udv + vdu ⇒ udv = uv − vdu.
For example, to integrate xex , we choose u = x and v = ex , so that du = dx and dv = ex , and
calculate
Z
Z
0
(22)
xex dx = xex + C − ex dx = xex − ex + C