Properties of the Sine Function y = sin(x)

Vocabulary flashcards summarizing the main concepts, properties, domain, range, periodicity, monotonic intervals, extrema, and parity of the function y = sin(x).

Domain of Definition of $y = \sin(x)$

The set of all real values for the argument (rotation angle) $x$, defined as $D(y) = (-\infty; +\infty)$.

Range of $y = \sin(x)$

The set of output values $E(y) = [-1; 1]$, bounded between $-1$ and $1$ because the radius of the unit circle is $1$ unit long.

Periodicity of $y = \sin(x)$

The property where the function repeats every period $T = 360^\circ = 2\pi$, given by the identity $\sin(x + 360^\circ) = \sin(x + 2\pi) = \sin(x)$.

Monotonic Function

A function whose values in a given interval only increase, only decrease, or remain unchanged as the argument value increases.

Intervals of Increase for $y = \sin(x)$

The argument intervals where the function values strictly increase, given by $x \in \left(-\frac{\pi}{2} + 2\pi k; \frac{\pi}{2} + 2\pi k\right)$, where $k \in \mathbb{Z}$.

Intervals of Decrease for $y = \sin(x)$

The argument intervals where the function values strictly decrease, given by $x \in \left(\frac{\pi}{2} + 2\pi k; \frac{3\pi}{2} + 2\pi k\right)$, where $k \in \mathbb{Z}$.

Positive Value Intervals of $y = \sin(x)$

The argument intervals where the sine function values are positive, given by $x \in (0 + 2\pi k; \pi + 2\pi k)$, where $k \in \mathbb{Z}$.

Negative Value Intervals of $y = \sin(x)$

The argument intervals where the sine function values are negative, given by $x \in (-\pi + 2\pi k; 0 + 2\pi k)$, where $k \in \mathbb{Z}$.

Maximum Value of $y = \sin(x)$

The largest value of the function is $1$, which is periodically reached at points $x = \frac{\pi}{2} + 2\pi k$, where $k \in \mathbb{Z}$.

Minimum Value of $y = \sin(x)$

The smallest value of the function is $-1$, which is periodically reached at points $x = -\frac{\pi}{2} + 2\pi k$, where $k \in \mathbb{Z}$.

Odd Function Property of $y = \sin(x)$

The parity property stating that $\sin(-x) = -\sin(x)$, demonstrating that $f(-x) = -f(x)$.

Sinusoid

The graph of the function $y = \sin(x)$, which is symmetric with respect to the origin of the coordinate plane $(0; 0)$ as an odd function graph.