Trigonometric

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oscillation

The Core Sine Function

We get the core sine function by viewing the $y$-coordinates of the unit circle as a function of the angle.

As you travel counterclockwise around the circle from the point $(1,0)$, the $y$-coordinate of the points

increase from $0$ to $1$, then
decrease from $1$ to $0$ and then to $-1$, then
increase from $-1$ to $0$

This keeps repeating with each full circle, counterclockwise (positive angles) or clockwise (negative angles).

The general sine function will be seen as compositions of the core sine function and two linear functions.

Sine Analysis

Domain: The domain of the core sine function is all real numbers, $(-\infty , \infty )$.

Zeros: The zeros of the core sine function are $\{ k\pi \, | \, k \in \mathbb {Z} \}$.

Continuity: The core sine function is continuous.

End-Behavior: The core sine function keeps oscillating between $1$ and $-1$.

Behavior (Increasing and Decreasing): The core sine function switches between increasing and decreasing with a period of $2\pi $.

On the interval $[0 , 2\pi )$, the Core sine function

increases on $\left (0, \frac {\pi }{2} \right )$
increases on $\left (\frac {\pi }{2}, \frac {3\pi }{2}\right )$
increases on $\left (\frac {3\pi }{2}, 2\pi \right )$

Global Maximum and Minimum:

The global maximum value is $1$, which occurs at $\{ \frac {\pi }{2} + 2k\pi \, | \, k \in \mathbb {Z} \}$.

The global minimum value is $-1$, which occurs at $\{ \frac {3\pi }{2} + 2k\pi \, | \, k \in \mathbb {Z} \}$.

Local Maximums and Minimums: The global maximum and minimum are the only local extrema.

Range: The range of the core sine function is $[-1, 1]$.

The Core Cosine Function

We get the core cosine function by viewing the $x$-coordinates of the unit circle as a function of the angle.

As you travel counterclockwise around the circle from the point $(1,0)$, the $x$-coordinate of the points

decrease from $1$ to $0$ and then to $-1$, then
increase from $-1$ to $0$ and then to $1$

This keeps repeating with each full circle, counterclockwise (positive angles) or clockwise (negative angles).

The general cosine function will be seen as compositions of the core cosine function and two linear functions.

Cosine Analysis

Domain: The domain of the core cosine function is all real numbers, $(-\infty , \infty )$.

Zeros: The zeros of the core sine function are $\{ \frac {\pi }{2} + k\pi \, | \, k \in \mathbb {Z} \}$.

Continuity: The core cosine function is continuous.

End-Behavior: The core cosine function keeps oscillating between $1$ and $-1$.

Behavior (Increasing and Decreasing): The core cosine function switches between increasing and decreasing with a period of $2\pi $.

On the interval $[0 , 2\pi )$, the core cosine function

decreases on $(0, \pi )$
increases on $(\pi , 2\pi )$

Global Maximum and Minimum:

The global maximum value is $1$, which occurs at $\{ 2k\pi \, | \, k \in \mathbb {Z} \}$.

The global minimum value is $-1$, which occurs at $\{ (2k+1)\pi \, | \, k \in \mathbb {Z} \}$.

Local Maximums and Minimums: The global maximum and minimum are the only local extrema.

Range: The range of the core cosine function is $[-1, 1]$.

The Core Tangent Function

Rational functions were quotients of polynomials. We can also create quotients with trigonometric functions.

Tangent

Tangent is the quotient of sine and cosine

\[ \tan (\theta ) = \frac {\sin (\theta )}{\cos (\theta )} \]

Its domain is all real numbers except the zeros of cosine, $\left \{ \frac {(2k+1)\pi }{2} \, | \, k \in \textbf {Z} \right \}$.

At each of these singularities, the graph has a vertical asymptote, where the function becomes unbounded.

Graph of $z = \tan (\theta )$.

Tangent takes on every value in $\mathbb {R}$ exactly once inside the open interval $\left ( \frac {-\pi }{2}, \frac {\pi }{2} \right )$. It is then periodic with a period of $\pi $.

Tangent is always increasing decreasing inside its domain.

Tangent is unbounded and has no global or local maximums or minimums.

Tangent has an infinite number of zeros. They coincide with the zeros of sine cosine : $\{ k \pi \, | \, k \in \textbf {Z} \}$

Tangent Analysis

Domain: The domain of the core tangent function is all real, except the zeros of cosine.

\[ (-\infty , \infty ) - \{ \frac {\pi }{2} + k\pi \, | \, k \in \mathbb {Z} \} \]

Another way to say this is the domain is the union of all intervals of the form

\[ \left ( -\frac {\pi }{2} + k\pi , \frac {\pi }{2} + k\pi \right ) \, | \, k \in \mathbb {Z} \]

These real numbers left out of the domain are singularities of tangent.

Zeros: The zeros of the core tangent are the zeros of the Core sine function, which are $\{ \frac {\pi }{2} + k\pi \, | \, k \in \mathbb {Z} \}$.

Continuity: The core tangent function is continuous.

End-Behavior: The core tangent function keeps repeating with a period of $\pi $.

It does become unbounded near its singularities.

\[ \lim \limits _{\theta \to -\frac {\pi }{2}^+} \tan (\theta ) = -\infty \]

\[ \lim \limits _{\theta \to \frac {\pi }{2}^-} \tan (\theta ) = \infty \]

Behavior (Increasing and Decreasing): The core tangent function is an increasing function.

Global Maximum and Minimum: The core tangent function has no global maximum or minimum.

Local Maximums and Minimums: The core tangent function has no local maximums or minimums.

Range: The range of the core cosine function is $[-1, 1]$.

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2026-05-30 01:34:24