Proof of The Least Positive Zero of the Cosine

Routine facts about $\mathbb{R}$ are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity); results stated for Derivative of a Real Function on an Interval (ii) and Derivative of a Real Function on an Interval (iii) are read for Derivative of a Real Function on an Interval (ii) and Derivative of a Real Function on an Interval (iii) by The Earlier and the New Versions of the Definitions of the Derivative on an Interval and of the Riemann Integral Agree (ii) and The Earlier and the New Versions of the Definitions of the Derivative on an Interval and of the Riemann Integral Agree (iii). Three elementary inequalities, obtained by comparing derivatives, show that the cosine becomes negative before $2$; the intermediate value theorem then produces a zero, and the least one is found as an infimum. Throughout we use $\cos 0=1$ and $\sin 0=0$, from Values at Zero, Parity and Double-Angle Identities for Sine and Cosine (i).

Three inequalities. We first show that for every real $t\ge0$

$$\sin t\le t,\qquad \cos t\ge1-\frac{t^{2}}{2},\qquad \sin t\ge t-\frac{t^{3}}{6}.$$

To this end consider, on the ray $[0,\infty)$ (The Real Line: Standing Notation and Background for Calculus on Intervals (v)), which contains the two distinct points $0$ and $1$, the three functions

$$f_{1}(t)=t-\sin t,\qquad f_{2}(t)=\cos t-1+\frac{t^{2}}{2},\qquad f_{3}(t)=\sin t-t+\frac{t^{3}}{6},$$

each of which vanishes at $t=0$. By Continuity, Derivatives and Power Series of Sine and Cosine (ii), $\sin$ and $\cos$ are differentiable on $\mathbb{R}$ with $\sin'=\cos$ and $\cos'=-\sin$; so their restrictions to $[0,\infty)$ are differentiable at every point of $[0,\infty)$, with the same derivatives, by Restriction Stability of Continuity and of the Derivative (ii). The remaining terms are handled by the rules for constants, for $t\mapsto t$, for powers, for sums and for scalar multiples (Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (i), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (ii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iii), Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (iv) and Derivatives of Constants, the Identity, Sums, Scalar Multiples, Products, Reciprocals, Quotients and Powers (v)): thus each $f_{k}$ is differentiable at every point of $[0,\infty)$, hence continuous on $[0,\infty)$ by The Increment Form of the Derivative, and Differentiability Implies Continuity (ii) and The Real Line: Standing Notation and Background for Calculus on Intervals (vi), and its derivative is

$$f_{1}'(t)=1-\cos t,\qquad f_{2}'(t)=t-\sin t=f_{1}(t),\qquad f_{3}'(t)=\cos t-1+\frac{t^{2}}{2}=f_{2}(t).$$

The three inequalities now follow one from the other. Since $\cos t\le|\cos t|\le1$ by The Pythagorean Identity for Sine and Cosine (ii), we have $f_{1}'\ge0$ on $[0,\infty)$, and so $f_{1}$ is nondecreasing there by The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds (i); that is, $f_{1}(t)\ge f_{1}(0)=0$ for $t\ge0$, which is the first inequality. Consequently $f_{2}'=f_{1}\ge0$, and the same clause gives $f_{2}(t)\ge f_{2}(0)=0$, the second inequality. Consequently, again, $f_{3}'=f_{2}\ge0$, and the same clause gives $f_{3}(t)\ge f_{3}(0)=0$, the third.

The cosine is negative at $2$. Taking $t=1$ in the third inequality gives $\sin1\ge1-\frac16=\frac{5}{6}$, so $\sin 1$ is positive and $(\sin1)^{2}\ge\frac{25}{36}$. The double-angle formula Values at Zero, Parity and Double-Angle Identities for Sine and Cosine (iii), with $x=1$, therefore gives

$$\cos2=1-2(\sin1)^{2}\le1-\frac{50}{36}=-\frac{7}{18}<0 .$$

The least zero. Let

$$Z=\{t\in[0,2]:\cos t=0\}.$$

The cosine is continuous on $\mathbb{R}$ by Continuity, Derivatives and Power Series of Sine and Cosine (i), hence its restriction to $[0,2]$ is continuous on $[0,2]$ (The Real Line: Standing Notation and Background for Calculus on Intervals (vi)); since $\cos0=1>0>\cos2$, the intermediate value theorem Intermediate Value Theorem on a Closed Real Interval, and Continuous Images of Intervals (i), applied with $y=0$, provides an element of $Z$. Thus $Z$ is nonempty and bounded below by $0$, so it has an infimum, which we call $x_{0}$; as $0$ bounds $Z$ from below and every element of $Z$ is at most $2$, we have $0\le x_{0}\le2$. We show that $x_{0}$ has the three properties of (i), and that it is the only number having them.

We shall use the following observation twice: if $0\le s\le2$ and $\cos s<0$, then $Z$ has an element smaller than $s$. Indeed, $s\neq0$ because $\cos0=1$, so $s>0$; the restriction of the cosine to $[0,s]$ is continuous, with $\cos0>0>\cos s$, so Intermediate Value Theorem on a Closed Real Interval, and Continuous Images of Intervals (i) gives $c\in[0,s]$ with $\cos c=0$. Then $c\neq s$, since $\cos s\neq0$; hence $c<s\le2$, and $c$ is an element of $Z$ smaller than $s$.

Next, $\cos x_{0}=0$. Suppose first $\cos x_{0}<0$: then the observation, with $s=x_{0}$, gives an element of $Z$ smaller than the infimum $x_{0}$ of $Z$, which is impossible. Suppose next $\cos x_{0}>0$. We first choose $\varepsilon=\cos x_{0}>0$; the continuity of the cosine at $x_{0}$, in its $\varepsilon$-$\delta$ form (The Real Line: Standing Notation and Background for Calculus on Intervals (vi)), then provides a $\delta>0$ such that $|\cos t-\cos x_{0}|<\cos x_{0}$, and therefore $\cos t>0$, for every real $t$ with $|t-x_{0}|<\delta$. Now, since $x_{0}$ is the infimum of $Z$, the larger number $x_{0}+\delta$ does not bound $Z$ from below, so (The Real Line: Standing Notation and Background for Calculus on Intervals (iii)) there is an element $z\in Z$ with

$$x_{0}\le z<x_{0}+\delta .$$

For this $z$ we have $|z-x_{0}|<\delta$, hence $\cos z>0$, contradicting $z\in Z$. Both cases being impossible, $\cos x_{0}=0$. In particular $x_{0}\neq0$ and $x_{0}\neq2$, because $\cos0=1$ and $\cos2<0$; so

$$0<x_{0}<2 .$$

Further, $\cos t>0$ whenever $0\le t<x_{0}$. Such a $t$ lies in $[0,2]$, as $x_{0}\le 2$, and lies below the infimum of $Z$, so $t\notin Z$, that is $\cos t\neq0$. If we had $\cos t<0$, the observation, with $s=t$, would give an element of $Z$ smaller than $t$, hence smaller than $x_{0}$, which is again impossible. Hence $\cos t>0$.

Finally, $x_{0}$ is the only number with these properties. Let $x_{1}$ be a real number with $0<x_{1}$, $\cos x_{1}=0$, and $\cos t>0$ for $0\le t<x_{1}$. If $x_{1}<x_{0}$, then $0\le x_{1}<x_{0}$, so $\cos x_{1}>0$ by the last property of $x_{0}$, contradicting $\cos x_{1}=0$; if $x_{0}<x_{1}$, the same argument with the roles of $x_{0}$ and $x_{1}$ exchanged gives $\cos x_{0}>0$, again a contradiction. Hence $x_{1}=x_{0}$. This proves (i), including the bound $x_{0}<2$.

The sine up to $x_{0}$. The interval $[0,x_{0}]$ contains the two distinct points $0$ and $x_{0}$. On it the restriction of the sine is continuous, by Continuity, Derivatives and Power Series of Sine and Cosine (i) and The Real Line: Standing Notation and Background for Calculus on Intervals (vi), and differentiable at every point, with derivative $\cos t$ at $t$, by Continuity, Derivatives and Power Series of Sine and Cosine (ii) and Restriction Stability of Continuity and of the Derivative (ii). At every inner point $t$ of $[0,x_{0}]$, that is for $0<t<x_{0}$, this derivative is positive by (i). Hence, by The Sign and Size of the Derivative: Monotonicity, Constancy and Lipschitz Bounds (ii), $\sin s<\sin t$ whenever $0\le s<t\le x_{0}$; taking $s=0$,

$$\sin t>\sin 0=0\qquad\text{for }0<t\le x_{0}.$$

Finally, the Pythagorean identity The Pythagorean Identity for Sine and Cosine (i) and $\cos x_{0}=0$ give

$$(\sin x_{0})^{2}=1-(\cos x_{0})^{2}=1,$$

and since $\sin x_{0}>0$, it follows that $\sin x_{0}=1$. This proves (ii).