Sine Function

Introduction

1.The geometric construction of the sine from the unit circle is developed in sine and cosine. Here the sine is treated as a real function of a real variable.

2.The sine function $f(x) = \sin(x)$ assigns to each angle $x,$ measured in radians, its corresponding sine value. Its graph is a periodic wave with period $2\pi$ and amplitude $1,$ oscillating between $-1$ and $1.$ The function has all real numbers in its domain, and its range is the interval $[-1, 1].$

3.Together with the cosine function, the sine models periodic phenomena. In simple harmonic motion the displacement of a mass on a spring or of a pendulum varies sinusoidally with time, and the velocity, as its derivative, is sinusoidal as well.

Properties

4.The following properties of the sine function follow from its definition on the unit circle.

Limits, derivatives, and integrals of the sine function

6.A remarkable limit involving the sine function describes its behaviour in a neighbourhood of the origin and is used to compute its derivative. As $x$ approaches zero, $\sin(x)$ becomes increasingly close to $x$ when both are measured in radians, so near the origin the sine curve is almost indistinguishable from the line $y = x.$ This is expressed by:

$$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$$

7.The function $\sin(x)$ is continuous and differentiable for every real value of $x.$ Its derivative is:

$$\frac{d}{dx}\sin(x) = \cos(x)$$

8.Differentiating again, each successive derivative follows from the previous one, and after four steps the function returns to itself:

$$ \begin{align} \frac{d^2}{dx^2}\sin(x) &= -\sin(x) \\[6pt] \frac{d^3}{dx^3}\sin(x) &= -\cos(x) \\[6pt] \frac{d^4}{dx^4}\sin(x) &= \sin(x) \end{align} $$

9.The derivatives repeat with period four, so the $n$-th derivative has the closed form:

$$\frac{d^n}{dx^n}\sin(x) = \sin\left(x + \frac{n\pi}{2}\right)$$

10.Since the derivative of $-\cos(x)$ is $\sin(x),$ the indefinite integral of the sine function is:

$$\int \sin(x) \ dx = -\cos(x) + c$$

11.

A broader treatment of trigonometric integrals, with the transformation and substitution techniques for the more complex cases, is given in trigonometric function integrals.

12.The sine function can also be written using imaginary numbers. With $e^{ix}$ the exponential function of base $e$ and $i$ the imaginary unit, Euler's formula gives:

$$\sin(x) = \frac{e^{ix} - e^{-ix}}{2i}$$

Monotonicity and convexity

13.The sine function increases where the derivative $\cos(x)$ is positive and decreases where it is negative. Within each period the intervals of increase and decrease are:

$$ \begin{align} \text{increase:} \quad & \left(-\frac{\pi}{2} + 2k\pi, \frac{\pi}{2} + 2k\pi\right) \\[6pt] \text{decrease:} \quad & \left(\frac{\pi}{2} + 2k\pi, \frac{3\pi}{2} + 2k\pi\right) \end{align} $$

14.In both cases $k \in \mathbb{Z},$ and the endpoints of these intervals are the maximum and minimum points.

15.The second derivative determines convexity and concavity. It is:

$$\frac{d^2}{dx^2}\sin(x) = -\sin(x)$$

16.The graph is concave where $\sin(x) > 0$ and convex where $\sin(x) < 0.$ The two behaviours meet at the inflection points $x = n\pi$ with $n \in \mathbb{Z},$ where the second derivative vanishes and changes sign. These coincide with the roots of the function.

Maclaurin series

17.The Maclaurin series of a function is its Taylor series centred at the origin, a power series whose partial sums approximate the function near $x = 0$ and are used to compute its values and to evaluate limits and integrals. For the sine function the series converges for every real number:

$$\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$

18.Only odd powers appear, in agreement with the sine being an odd function. Keeping the first term gives the approximation $\sin(x) \approx x$ for small $x,$ which recovers the limit $\dfrac{\sin(x)}{x} \to 1$ as $x \to 0.$