Response graphs show a complete period; the dashed curve, when it appears, is the “reference” function ($\operatorname{sen} x$ or $\cos x$). To check your graph, also use $a + b\operatorname{sen}(cx + d)$ explorer. Sine function In exercises 23 to 42, determine the period and the image and plot the graph of a complete period of the given function $f\colon \mathbb{R} \to \mathbb{R}$. 23 $f(x) = -\operatorname{sen} x$. See the resolutionWe build a table in three steps: 1st) we assign values to $x$; 2nd) we associate the value of $\operatorname{sen} x$ to each $x$; 3rd) we multiply $\operatorname{sen} x$ by $-1$. $x$ $0$ $\frac{\pi}{2}$ $\pi$ $\frac{3\pi}{2}$ $2\pi$ $\operatorname{sen} x$ $0$ $1$ $0$ $-1$ $0$ $y$ $0$ $-1$ $0$ $1$ $0$ With this table we obtain 5 points of the graph, which is the symmetric of the sinusoid in relation to the $x$ axis. It is immediate that $\operatorname{Im}(f) = [-1, 1]$ and $p(f) = 2\pi$. Ative o JavaScript para ver esta figura interativa.
The sine is the height of the point that rotates in the cycle. When the point turns, the height rises and falls between $-1$ and $1$, drawing the most famous curve in mathematics: the sinusoid. The story For more than a thousand years, the sine was a table number: no one thought about drawing its graph. The first known sinusoid appeared around 1634, in a work by the Frenchman Gilles Personne de Roberval on the cycloid, the curve described by a point on a rolling wheel. To calculate the area under the cycloid, Roberval used an auxiliary curve, which he called the “companion of the cycloid” — and this companion is exactly a sinusoid. It was Euler, a century later, who began to treat the sine as a function of a real number, and only then did the graph gain the central place it has today.
Before defining the circular functions, we equipped the cycle with four axes and a language to say in which “piece” of the loop a number is. Then we study the property that all these functions will have in common: periodicity. The story The word sine has a curious history. Fifth-century Indian astronomers such as Aryabhata tabulated the half-chord, which they called jya-ardha or simply jya. Arabic translators transcribed the term as jiba, written only with consonants, jb. In the 12th century, Latin translators read jaib — “fold,” “breast” — and translated it as sinus. The tangent and secant gained their name from the Danish Thomas Fincke, in 1583, and the prefix co- of cosine, cotangent and cosecant, which means “of the complement”, was fixed by Edmund Gunter in 1620.