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LECTURE 11: MORE ON TRIGONOMETRIC FUNCTIONS
1. Further standard trigonometric functions
Having defined the functions cos : R Ñ r´1, 1s and sin : R Ñ r´1, 1s in the
previous lecture, one may easily construct other standard trigonometric functions.
Definition. The secant function sec : Rztpk ` 1{2qπ : k P Zu Ñ R and tangent
function tan : Rztpk ` 1{2qπ : k P Zu Ñ R are given by
sin x
1
,
tan x :“
for x P Rztpk ` 1{2qπ : k P Zu.
sec x :“
cos x
cos x
Definition. The cosecant function csc : Rztkπ : k P Zu Ñ R and cotangent function cot : Rztkπ : k P Zu Ñ R are given by
cos x
1
,
cot x :“
for x P Rztkπ : k P Zu.
csc x :“
sin x
sin x
By using the usual rules for differentiation and the established formulae for cos1
and sin1 , one may easily prove the following theorem.
Theorem. For x P Rztpk ` 1{2qπ : k P Zu one has
sec1 pxq “ secpxq tanpxq,
tan1 pxq “ sec2 pxq.
For x P Rztkπ : k P Zu one has
csc1 pxq “ ´ cscpxq cotpxq,
cot1 pxq “ ´ csc2 pxq.
It is, of course, also useful to study inverse trigonometric functions. Clearly sin
and cos are not bijective on the whole of R and so one must restrict the domain to
ensure the inverses are defined. A standard choice of (maximal) domains is given
by
sin : r´π{2, π{2s Ñ r´1, 1s,
cos : r0, πs Ñ r´1, 1s,
tan : p´π{2, π{2q Ñ R.
Each of these functions can easily be seen to be bijective (for instance, the derivatives of sin and tan are strictly positive on p´π{2, π{2q).
Definition.
‚ The inverse of cos : r0, πs Ñ r´1, 1s is denoted arccos. Hence
arccos : r´1, 1s Ñ r0, πs satisfies
arccospcos xq “ x
for all x P r0, πs.
‚ The inverse of sin : r´π{2, π{2s Ñ r´1, 1s is denoted arcsin. Hence arcsin : r´1, 1s Ñ
r´π{2, π{2s satisfies
arcsinpsin xq “ x
for all x P r´π{2, π{2s.
‚ The inverse of tan : p´π{2, π{2q Ñ r´1, 1s is denoted arctan. Hence arctan : R Ñ
p´π{2, π{2q satisfies
arctanptan xq “ x
for all x P p´π{2, π{2q.
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LECTURE 11: MORE ON TRIGONOMETRIC FUNCTIONS
Recall that we defined cos to be the inverse of twice the area function A in the
last lecture. Hence, recalling the integral formula for A, it follows that
ż1a
a
arccospxq “ 2Apxq “ x 1 ´ x2 ` 2
1 ´ t2 dt.
x
1
We also previously computed the derivative A on p´1, 1q and from our analysis it
follows that
´1
arccos1 pxq “ ?
for x P p´1, 1q.
1 ´ x2
Since sin1 and tan1 are strictly positive on p´π{2, π{2q, it follows from the theory of
inverse functions that arcsin is differentiable on p´1, 1q and arctan is differentiable
on R.
Theorem. If x P p´1, 1q, then
´1
,
1 ´ x2
1
arcsin1 pxq “ ?
.
1 ´ x2
arccos1 pxq “ ?
If x P R, then
arctan1 pxq “
1
.
1 ` x2
2. The double angle formula
We conclude this discussion of trigonometric functions with a calculus-based
proof of the familiar double angle formulae.
Theorem. For x, y P R one has
cospx ` yq “ cos x cos y ´ sin x sin y,
sinpx ` yq “ sin x cos y ` cos x sin y.
Our proof is based on elementary theory of differential equations.
Lemma. Suppose f : R Ñ R is everywhere twice differentiable and that
f 2 ` f “ 0,
f p0q “ 0,
f 1 p0q “ 0.
Then f “ 0.
Proof. Multiplying the equation through by 2f 1 it follows that
pf 12 ` f 2 q1 “ 2f 2 f 1 ` 2f 1 f “ 0
and therefore f 12 ` f 2 is constant. The conditions f p0q “ f 1 p0q “ 0 imply that
f 12 ` f 2 “ 0 and so f “ 0.
This lemma is a special case of the following more general theorem.
Theorem. Suppose f : R Ñ R is everywhere twice differentiable and that
f 2 ` f “ 0,
f p0q “ a,
f 1 p0q “ b.
Then f “ a cos `b sin.
LECTURE 11: MORE ON TRIGONOMETRIC FUNCTIONS
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Jonathan Hickman, Department of mathematics, University of Chicago, 5734 S. University Avenue, Eckhart hall Room 414, Chicago, Illinois, 60637.
E-mail address: [email protected]