Sine

sine and cosine, abbreviated sin and cos, are basic trigonometric functions (which are, in their turn, basic elementary functions), used to express relations between angles and sides of right triangles). These functions are common in science and technology.

Various sources define trigonometric funcitons in different ways.

Geometric approach

In some countries, schoolchildren begin to study mathematics with arithmetics, elementary algebra and the Euclidean geometry (planimetry) .

Beginning to deal with sin and cos, students already have some idea about "addition" and measurement of angles, segments and areas; they already have accepted the axioms of planimetry, in particular, the Euclid's parallel axiom, and they know the Pythagorean theorem. Therefore, the trigonometric functions are defined as ratios of the lengths of sides of the right-angled triangles, for example, ratio of length of a leg to that of the hypothenuse.

The geometrical approach rather assumes properties of sin and cos than deduce them. The assumptions are masked as the geometric axioms. However the deduction of properties of trigonometric functions does not require axioms of geometry. Properties of sin and cos do not depend, for example, on the acceptance of rejection of the Euclid's parallel axiom. Funcitons sin and cos can be defined algebraically, then their properties can be deduced, not postulated. Basic elements of such a deduction are suggested below.

Definition by differential equations

The definitions of derivative and integral do not refer to trigonometric functions, and the basic properties of integrals and derivatives (in particular the Cauchy-Kowalevski theorem of existence and uniqueness of solution of the Cauchy problem) can be deduced without reference to sin, cos, or any axioms of Euclidean geometry. Therefore, a natural and consistent way of defining the functions sin and cos is via differential equations. We will indicate the first derivative of a function with respect to its variable by a prime.

Definition

The functions sin and cos are solutions of the following system of equations

(1)    $\sin '(t)=~\cos(t)\,$

(2)    $\cos '(t)=-\sin(t)\,$

with conditions

(3)    $\sin(0)=0\,$

(4)    $\cos(0)=1\,$

Usually, it is assumed that the independent variable t is real; in that case, the values of the functions sin and cos are also real numbers. However, the definition above can be used for complex numbers too.

The present definition does not imply additional concepts about angles and sums of angles; and does not require the Pythagorean theorem or even the parallel axiom of Euclidean geometry. Furthermore the definition does not refer to the concept of the number Pi ($\,\pi$) and therefore can be used for the definition of $\,\pi$. But, one pays for this by having to deduce the naively-obvious properties of sin and cos from the system (1)–(4).

Sum of squares

Consider the function $\,z(t)=\sin(t)^{2}+\cos(t)^{2}$. From equations (1) and (2) it follows that $\,z'(t)=0$, i.e., $\,z(t)=z(0)=$constant . Evaluation of this constant from equations (2) and (3) gives $\,z(t)=1$ for all $\,t$, i.e.,

(5)    $\,\sin(t)^{2}+\cos(t)^{2}=1$

In this article, the superscript as indicator of exponentiation is always written last to specify the argument, avoiding the confusions.

For the equation (5) it follows, that for all real values of t the functions sin(t) and cos(t) are bounded:

(6)    $-1\leq \sin(t)\leq 1$

(7)    $-1\leq \cos(t)\leq 1$

These properties allow efficient bracketing of functions sin and cos.

Bracketing

Consider $sin(x)$ and $\cos(x)$ in the case $x>0$. From equation (1) it follows, that $\sin '(x)\leq 1$. Integration of the last inequality with respect to $x$ from 0 to $t$ gives: $\sin(t)-\sin(0)<t$. Using equation (3) we write

(8) $\sin(t)<t$ .

Rewrite (8) as $-t<-\sin(t)$ and apply equation(2):

$-t<\cos '(t)$

Integrating this inequality with respect to $t$ from 0 to $x$ gives:

$-x^{2}/2<\cos(x)-\cos(0)$

Using equation(4) gives

(9) $1-x^{2}/2<\cos(x)<1$

The last "<" just follows from equation (7). Using equation (1) gives

(10) $1-x^{2}/2<\sin '(x)<1$

Integrating equation (12) with respect to $x$ from 0 to $t$ gives

(11) $t-t^{3}/6<\sin(t)<t$

Rewrite it as

$-t<-\sin(t)<-t+t^{3}/6$

and apply equation (2):

$-t<\cos '(t)<-t+t^{3}/6$

Integrating this equation with respect to $t$ from 0 to $x$ gives

$-x^{2}/2<\cos(x)-\cos(0)<-x^{2}/2+x^{3}/6$

(12) $1-x^{2}/2<\cos(x)<1-x^{2}/2+x^{4}/24$

Continuing this exercise, one can obtain more and more narrow bounds for values of sin and cos.

However, even equations (9,12) are sufficient to see that both both sin and cos remain positive while, the argument is within (0,1) range. This property is used below to reveal periodicity of these funcitons.

Mathematical induction

Applying the mathematical induction, it is possible to show that for positive integer $N$

(13) $\sum _{n=0}^{2N+1}{\frac {x~(-x^{2})^{n}}{(2n+1)!}}\leq \sin(x)\leq \sum _{n=0}^{2N}{\frac {x(-x^{2})^{n}}{(2n+1)!}}$

(14) $\sum _{n=0}^{2N+1}{\frac {~~(-x^{2})^{n}}{(2n)!}}\leq \cos(x)\leq \sum _{n=0}^{2N}{\frac {(-x^{2})^{n}}{(2n)!}}$

and equality takes place only at $x=0$.

Is cos always larger than sin?

Consider the range $0<t<P$ such that both $\sin(t)$ and $\cos(t)$ are positive; let $P$ be exact upper bound of such interval. Within this interval, sin is monotonously increasing, and cos is monotonously decreasing. Therefore, there exist one and only one solution $p$ of the equation

(13) $\sin(p)=\cos(p)$

Let us estimate value of this solution.

Consider the special case, substitute $x=1$ into equation(13) and $t=1$ into equation(11). This gives

$\cos(1)<1-1/2+1/24=13/24$

$\sin(1)>1-1/6=5/6=20/24$

In such a way,

(14) $\sin(1)>\cos(1)$

hence,

(15) $p<1$.

Substitute $x=3/4$ into equation (11); this gives

(16) $\cos(3/4)>1-(3/4)^{2}/2=23/32$

(17) $\cos(3/4)^{2}>529/1024>1/2$

From equation (5) it follows that

(18) $\sin(3/4)^{2}=1-\cos(3/4)^{2}<1-529/1024<1/2$

(19) $\sin(3/4)^{2}<\cos(3/4)^{2}$

Both sin and cos are positive in the range (0,1); and, therefore,

(20) $\sin(3/4)<\cos(3/4)$

Therefore,

(21) $3/4<p<1$.

Symmetry and sense of number $P$

Change of variables $t$ to $-t$ and replacement $\sin$ to $-\sin$ keeps the system (1)-(4) invariant. This means that

(22) $\sin(-t)=-\sin(t)$

(23) $\cos(-t)=\cos(t)$

Consider replacement $t$ to $p-t$, cos to sin, and sin to cos. Due to equation (13), such a replacement preserves the equations (1-4); therefore,

(24) $\sin(p-t)=\cos(t)$

(25) $\cos(t-p)=\sin(t)$

Then, $P=2p$, and

(26) $\sin(t+P)=\cos(P)$

(27) $\cos(t+P)=-\sin(P)$

It follows, that functions sin and cos are periodic; and

(30) $2\pi =4P=8p$

is period. The deduction above can be used as definition of number $\pi$. In such a way, $\pi$ appears as half of period of functions which are solutions of equations (1,2).

For definition of $\pi$, equations (1,2) are sufficient: due to the linearity of equations and the translational invariance, conditions (3,4) do not affect the period; all the solutions with different conditions at zero have the same period.

From estimate (21) it follows that

(31)$3<\pi <4$

The improvement of this estimate using the expansions (13), (14) is possible but computationally non-efficient. The efficient algorithms for evaluation of $\pi$ are mentioned in the spacial article about Pi. Usually, the efficient algorithms require more efforts for the deduction.