1.1: Trigonometric Functions

Last updated

Dec 12, 2023
Save as PDF
- Chapter 1: Functions and Graphs
- Chapter 2: Limits

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Express $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ in terms of $\sin(\theta)$ and $\cos(\theta)$ .
Convert $180^\circ$ from degrees to radians.
Convert $\dfrac{\pi}{4}$ from radians to degrees.
Convert $270^\circ$ from degrees to radians.
Convert $\dfrac{\pi}{6}$ from radians to degrees.
Convert $60^\circ$ from degrees to radians.
Convert $\dfrac{\pi}{2}$ from radians to degrees.
Given the triangle below, find two sets of values for the lengths of the sides $a$ , $b$ , and $c$ .
$A right triangle with an angle equal to pi/6. The side adjacent to this angle is labeled a, the side opposite this angle is labeled b, and the hypothenuse is labeled c.$
Given the triangle below, find two sets of values for the lengths of the sides $a$ , $b$ , and $c$ .
$A right triangle with an angle equal to pi/4. The side adjacent to this angle is labeled a, the side opposite this angle is labeled b, and the hypothenuse is labeled c.$
Given the triangle below, find two sets of values for the lengths of the sides $a$ , $b$ , and $c$ .
$A right triangle with an angle equal to pi/3. The side adjacent to this angle is labeled a, the side opposite this angle is labeled b, and the hypothenuse is labeled c.$
Given the triangle below, find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
$A right triangle with acute angle theta, whete the side adjacent to theta is 2, the side opposite theta is sqrt(5), and the hypotenuse is 3.$
Find the reference angle to $\dfrac{\pi}{3}$ .
Find the reference angle to $\dfrac{7\pi}{6}$ .
Find the reference angle to $\dfrac{3\pi}{4}$ .
Find the reference angle to $\dfrac{11\pi}{6}$ .
Find the reference angle to $\dfrac{5\pi}{4}$ .
Find the reference angle to $\dfrac{2\pi}{3}$ .
Find the reference angle to $\dfrac{9\pi}{4}$ .
Find the reference angle to $-\dfrac{2\pi}{3}$ .
For what values of $\theta \in [0, 2\pi]$ is $\sin \theta = 0$ ? For what values of $\theta \in [0, 2\pi]$ is $\sin \theta > 0$ ? For what values of $\theta \in [0, 2\pi]$ is $\sin \theta < 0$ ?
For what values of $\theta \in [0, 2\pi]$ is $\cos \theta = 0$ ? For what values of $\theta \in [0, 2\pi]$ is $\cos \theta > 0$ ? For what values of $\theta \in [0, 2\pi]$ is $\cos \theta < 0$ ?
For what values of $\theta \in [0, 2\pi]$ is $\tan \theta = 0$ ? For what values of $\theta \in [0, 2\pi]$ is $\tan \theta > 0$ ? For what values of $\theta \in [0, 2\pi]$ is $\tan \theta < 0$ ?
Given $\theta = 0$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = \dfrac{\pi}{3}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = \dfrac{3\pi}{4}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = \dfrac{\pi}{2}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = 2\pi$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = \dfrac{7\pi}{6}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = \dfrac{4\pi}{3}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = \dfrac{3\pi}{2}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = \dfrac{5\pi}{3}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = \pi$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = \dfrac{8\pi}{3}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = \dfrac{5\pi}{2}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = -\dfrac{\pi}{4}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = -\pi$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = -\dfrac{\pi}{2}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Given $\theta = -\dfrac{8\pi}{3}$ , find $\sin(\theta)$ , $\cos(\theta)$ , $\sec(\theta)$ , $\tan(\theta)$ , $\csc(\theta)$ , and $\cot(\theta)$ .
Find all values of $\alpha \in [0, 2\pi)$ where $\sin(\alpha) = \dfrac{1}{\sqrt{2}}$ .
Find all values of $\alpha \in [0, 2\pi)$ where $\cos(\alpha) = 0$ .
Find all values of $\alpha \in [0, 2\pi)$ where $\tan(\alpha) = -\sqrt{3}$ .
Find all values of $\alpha \in [0, 2\pi)$ where $\sec(\alpha) = -2$ .
Find all values of $\alpha \in [0, 2\pi)$ where $\cot(\alpha) = -1$ .
Find all values of $\alpha \in [0, 2\pi)$ where $\csc(\alpha)$ is undefined.

Search

Text Color

Text Size

Margin Size

Font Type

Support Center

How can we help?