Trigonometry Flashcards

Vocabulary flashcards covering unit circle definitions, function definitions, standard trigonometric values, and elementary trigonometric formulas from Chapter 4: Trigonometry.

Trigonometric Circle

A circle of center $O$ and radius $1$ defined in a direct orthonormal coordinate frame $(O, \mathbf{i}, \mathbf{j})$.

Point Coordinates on the Trigonometric Circle

For a point $M$ on the trigonometric circle such that the oriented angle $(\mathbf{i}, \vec{OM})$ measures $x$ radians, the coordinates of $M$ in the frame $(O, \mathbf{i}, \mathbf{j})$ are $(\cos(x), \sin(x))$.

Cosine Function

The function defined as $\cos: \mathbb{R} \rightarrow \mathbb{R}$ where $x \mapsto \cos(x)$.

Sine Function

The function defined as $\sin: \mathbb{R} \rightarrow \mathbb{R}$ where $x \mapsto \sin(x)$.

Trigonometric Values at $0$ Radians

$\cos(0) = 1$ and $\sin(0) = 0$.

Trigonometric Values at $\frac{\pi}{6}$ Radians

$\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$ and $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.

Trigonometric Values at $\frac{\pi}{4}$ Radians

$\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$ and $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$.

Trigonometric Values at $\frac{\pi}{3}$ Radians

$\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$ and $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$.

Trigonometric Values at $\frac{\pi}{2}$ Radians

$\cos\left(\frac{\pi}{2}\right) = 0$ and $\sin\left(\frac{\pi}{2}\right) = 1$.

Pythagorean Trigonometric Identity

For all $x \in \mathbb{R}$, $\cos^2(x) + \sin^2(x) = 1$.

Negation Formulas ($-x$)

For all $x \in \mathbb{R}$, $\cos(-x) = \cos(x)$ and $\sin(-x) = -\sin(x)$.

Formulas for $\pi - x$

For all $x \in \mathbb{R}$, $\cos(\pi - x) = -\cos(x)$ and $\sin(\pi - x) = \sin(x)$.

Formulas for $\pi + x$

For all $x \in \mathbb{R}$, $\cos(\pi + x) = -\cos(x)$ and $\sin(\pi + x) = -\sin(x)$.

Formulas for $\frac{\pi}{2} - x$

For all $x \in \mathbb{R}$, $\cos\left(\frac{\pi}{2} - x\right) = \sin(x)$ and $\sin\left(\frac{\pi}{2} - x\right) = \cos(x)$.

Formulas for $\frac{\pi}{2} + x$

For all $x \in \mathbb{R}$, $\cos\left(\frac{\pi}{2} + x\right) = -\sin(x)$ and $\sin\left(\frac{\pi}{2} + x\right) = \cos(x)$.