Finding all six trig functions means finding \(\sin\theta\), \(\cos\theta\), \(\tan\theta\), \(\csc\theta\), \(\sec\theta\), and \(\cot\theta\). These values can come from the unit circle, a point on the terminal side, or one trig function together with quadrant information. This page practices using reference angles, signs by quadrant, reciprocal identities, and the Pythagorean theorem.
Practice Problems
Find all 6 trig functions
\(\textbf{1)}\) Find all 6 trig functions if \(\theta=180^{\circ}\) \(\sin180^{\circ}=0\,\,\,\,\,\,\cos180^{\circ}=-1\,\,\,\,\,\,\tan180^{\circ}=0\)
\(\csc180^{\circ}=\text{undefined}\,\,\,\,\,\,\sec180^{\circ}=-1\,\,\,\,\,\,\cot180^{\circ}=\text{undefined}\)
\(\,\,\,\,\,\,180^\circ\text{ is on the negative }x\text{-axis.}\)
\(\,\,\,\,\,\,\left(x,y\right)=\left(-1,0\right)\)
\(\,\,\,\,\,\,\sin\theta=y=0\)
\(\,\,\,\,\,\,\cos\theta=x=-1\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{0}{-1}=0\)
\(\,\,\,\,\,\,\csc\theta=\frac{1}{\sin\theta}=\frac{1}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\sec\theta=\frac{1}{\cos\theta}=\frac{1}{-1}=-1\)
\(\,\,\,\,\,\,\cot\theta=\frac{1}{\tan\theta}=\frac{1}{0}=\text{undefined}\)
\(\csc180^{\circ}=\text{undefined}\,\,\,\,\,\,\sec180^{\circ}=-1\,\,\,\,\,\,\cot180^{\circ}=\text{undefined}\)
\(\,\,\,\,\,\,180^\circ\text{ is on the negative }x\text{-axis.}\)
\(\,\,\,\,\,\,\left(x,y\right)=\left(-1,0\right)\)
\(\,\,\,\,\,\,\sin\theta=y=0\)
\(\,\,\,\,\,\,\cos\theta=x=-1\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{0}{-1}=0\)
\(\,\,\,\,\,\,\csc\theta=\frac{1}{\sin\theta}=\frac{1}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\sec\theta=\frac{1}{\cos\theta}=\frac{1}{-1}=-1\)
\(\,\,\,\,\,\,\cot\theta=\frac{1}{\tan\theta}=\frac{1}{0}=\text{undefined}\)
\(\textbf{2)}\) Find all 6 trig functions if \(\theta=330^{\circ}\) \(\sin330^{\circ}=-\frac{1}{2}\,\,\,\,\,\,\cos330^{\circ}=\frac{\sqrt{3}}{2}\,\,\,\,\,\,\tan330^{\circ}=-\frac{\sqrt{3}}{3}\)
\(\csc330^{\circ}=-2\,\,\,\,\,\,\sec330^{\circ}=\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\cot330^{\circ}=-\sqrt{3}\)
\(\,\,\,\,\,\,330^\circ\text{ has reference angle }30^\circ\text{ and is in Quadrant IV.}\)
\(\,\,\,\,\,\,\sin330^\circ=-\frac{1}{2}\)
\(\,\,\,\,\,\,\cos330^\circ=\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\tan330^\circ=\frac{\sin330^\circ}{\cos330^\circ}=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\csc330^\circ=\frac{1}{\sin330^\circ}=-2\)
\(\,\,\,\,\,\,\sec330^\circ=\frac{1}{\cos330^\circ}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\cot330^\circ=\frac{1}{\tan330^\circ}=-\sqrt{3}\)
\(\csc330^{\circ}=-2\,\,\,\,\,\,\sec330^{\circ}=\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\cot330^{\circ}=-\sqrt{3}\)
\(\,\,\,\,\,\,330^\circ\text{ has reference angle }30^\circ\text{ and is in Quadrant IV.}\)
\(\,\,\,\,\,\,\sin330^\circ=-\frac{1}{2}\)
\(\,\,\,\,\,\,\cos330^\circ=\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\tan330^\circ=\frac{\sin330^\circ}{\cos330^\circ}=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\csc330^\circ=\frac{1}{\sin330^\circ}=-2\)
\(\,\,\,\,\,\,\sec330^\circ=\frac{1}{\cos330^\circ}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\cot330^\circ=\frac{1}{\tan330^\circ}=-\sqrt{3}\)
\(\textbf{3)}\) Find all 6 trig functions if \(\theta=-135^{\circ}\) \(\sin(-135^{\circ})=-\frac{\sqrt{2}}{2}\,\,\,\,\,\,\cos(-135^{\circ})=-\frac{\sqrt{2}}{2}\,\,\,\,\,\,\tan(-135^{\circ})=1\)
\(\csc(-135^{\circ})=-\sqrt{2}\,\,\,\,\,\,\sec(-135^{\circ})=-\sqrt{2}\,\,\,\,\,\,\cot(-135^{\circ})=1\)
\(\,\,\,\,\,\,-135^\circ\text{ is coterminal with }225^\circ.\)
\(\,\,\,\,\,\,225^\circ\text{ has reference angle }45^\circ\text{ and is in Quadrant III.}\)
\(\,\,\,\,\,\,\sin\theta=-\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\cos\theta=-\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\tan\theta=\frac{\sin\theta}{\cos\theta}=1\)
\(\,\,\,\,\,\,\csc\theta=\frac{1}{\sin\theta}=-\sqrt{2}\)
\(\,\,\,\,\,\,\sec\theta=\frac{1}{\cos\theta}=-\sqrt{2}\)
\(\,\,\,\,\,\,\cot\theta=\frac{1}{\tan\theta}=1\)
\(\csc(-135^{\circ})=-\sqrt{2}\,\,\,\,\,\,\sec(-135^{\circ})=-\sqrt{2}\,\,\,\,\,\,\cot(-135^{\circ})=1\)
\(\,\,\,\,\,\,-135^\circ\text{ is coterminal with }225^\circ.\)
\(\,\,\,\,\,\,225^\circ\text{ has reference angle }45^\circ\text{ and is in Quadrant III.}\)
\(\,\,\,\,\,\,\sin\theta=-\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\cos\theta=-\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\tan\theta=\frac{\sin\theta}{\cos\theta}=1\)
\(\,\,\,\,\,\,\csc\theta=\frac{1}{\sin\theta}=-\sqrt{2}\)
\(\,\,\,\,\,\,\sec\theta=\frac{1}{\cos\theta}=-\sqrt{2}\)
\(\,\,\,\,\,\,\cot\theta=\frac{1}{\tan\theta}=1\)
\(\textbf{4)}\) Find all 6 trig functions if the terminal side contains the point \((3,4)\) \(\sin\theta=\frac{4}{5}\,\,\,\,\,\,\cos\theta=\frac{3}{5}\,\,\,\,\,\,\tan\theta=\frac{4}{3}\)
\(\csc\theta=\frac{5}{4}\,\,\,\,\,\,\sec\theta=\frac{5}{3}\,\,\,\,\,\,\cot\theta=\frac{3}{4}\)
\(\,\,\,\,\,\,x=3,\quad y=4\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{3^2+4^2}=5\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=\frac{4}{5}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=\frac{3}{5}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{4}{3}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=\frac{5}{4}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=\frac{5}{3}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{3}{4}\)
\(\csc\theta=\frac{5}{4}\,\,\,\,\,\,\sec\theta=\frac{5}{3}\,\,\,\,\,\,\cot\theta=\frac{3}{4}\)
\(\,\,\,\,\,\,x=3,\quad y=4\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{3^2+4^2}=5\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=\frac{4}{5}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=\frac{3}{5}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{4}{3}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=\frac{5}{4}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=\frac{5}{3}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{3}{4}\)
\(\textbf{5)}\) Find all 6 trig functions if the terminal side contains the point \((-5,-12)\) \(\sin\theta=-\frac{12}{13}\,\,\,\,\,\,\cos\theta=-\frac{5}{13}\,\,\,\,\,\,\tan\theta=\frac{12}{5}\)
\(\csc\theta=-\frac{13}{12}\,\,\,\,\,\,\sec\theta=-\frac{13}{5}\,\,\,\,\,\,\cot\theta=\frac{5}{12}\)
\(\,\,\,\,\,\,x=-5,\quad y=-12\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{(-5)^2+(-12)^2}=13\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-\frac{12}{13}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=-\frac{5}{13}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{-12}{-5}=\frac{12}{5}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-\frac{13}{12}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=-\frac{13}{5}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{-5}{-12}=\frac{5}{12}\)
\(\csc\theta=-\frac{13}{12}\,\,\,\,\,\,\sec\theta=-\frac{13}{5}\,\,\,\,\,\,\cot\theta=\frac{5}{12}\)
\(\,\,\,\,\,\,x=-5,\quad y=-12\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{(-5)^2+(-12)^2}=13\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-\frac{12}{13}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=-\frac{5}{13}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{-12}{-5}=\frac{12}{5}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-\frac{13}{12}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=-\frac{13}{5}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{-5}{-12}=\frac{5}{12}\)
\(\textbf{6)}\) Find all 6 trig functions if \(\sin(\theta)=-\frac{4}{5}\), Quadrant III \(\sin\theta=-\frac{4}{5}\,\,\,\,\,\,\cos\theta=-\frac{3}{5}\,\,\,\,\,\,\tan\theta=\frac{4}{3}\)
\(\csc\theta=-\frac{5}{4}\,\,\,\,\,\,\sec\theta=-\frac{5}{3}\,\,\,\,\,\,\cot\theta=\frac{3}{4}\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-\frac{4}{5}\)
\(\,\,\,\,\,\,y=-4,\quad r=5\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,x^2+(-4)^2=5^2\)
\(\,\,\,\,\,\,x^2+16=25\)
\(\,\,\,\,\,\,x^2=9\)
\(\,\,\,\,\,\,x=-3\text{ because Quadrant III has negative }x\text{ and negative }y.\)
\(\,\,\,\,\,\,\sin\theta=-\frac{4}{5},\quad \cos\theta=-\frac{3}{5},\quad \tan\theta=\frac{4}{3}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{5}{4},\quad \sec\theta=-\frac{5}{3},\quad \cot\theta=\frac{3}{4}\)
\(\csc\theta=-\frac{5}{4}\,\,\,\,\,\,\sec\theta=-\frac{5}{3}\,\,\,\,\,\,\cot\theta=\frac{3}{4}\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-\frac{4}{5}\)
\(\,\,\,\,\,\,y=-4,\quad r=5\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,x^2+(-4)^2=5^2\)
\(\,\,\,\,\,\,x^2+16=25\)
\(\,\,\,\,\,\,x^2=9\)
\(\,\,\,\,\,\,x=-3\text{ because Quadrant III has negative }x\text{ and negative }y.\)
\(\,\,\,\,\,\,\sin\theta=-\frac{4}{5},\quad \cos\theta=-\frac{3}{5},\quad \tan\theta=\frac{4}{3}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{5}{4},\quad \sec\theta=-\frac{5}{3},\quad \cot\theta=\frac{3}{4}\)
\(\textbf{7)}\) Find all 6 trig functions if \(\theta=90^\circ\) \(\sin90^\circ=1\,\,\,\,\,\,\cos90^\circ=0\,\,\,\,\,\,\tan90^\circ=\text{undefined}\)
\(\csc90^\circ=1\,\,\,\,\,\,\sec90^\circ=\text{undefined}\,\,\,\,\,\,\cot90^\circ=0\)
\(\,\,\,\,\,\,90^\circ\text{ is on the positive }y\text{-axis.}\)
\(\,\,\,\,\,\,\left(x,y\right)=\left(0,1\right)\)
\(\,\,\,\,\,\,\sin\theta=y=1\)
\(\,\,\,\,\,\,\cos\theta=x=0\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{1}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\csc\theta=\frac{1}{\sin\theta}=1\)
\(\,\,\,\,\,\,\sec\theta=\frac{1}{\cos\theta}=\frac{1}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{0}{1}=0\)
\(\csc90^\circ=1\,\,\,\,\,\,\sec90^\circ=\text{undefined}\,\,\,\,\,\,\cot90^\circ=0\)
\(\,\,\,\,\,\,90^\circ\text{ is on the positive }y\text{-axis.}\)
\(\,\,\,\,\,\,\left(x,y\right)=\left(0,1\right)\)
\(\,\,\,\,\,\,\sin\theta=y=1\)
\(\,\,\,\,\,\,\cos\theta=x=0\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{1}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\csc\theta=\frac{1}{\sin\theta}=1\)
\(\,\,\,\,\,\,\sec\theta=\frac{1}{\cos\theta}=\frac{1}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{0}{1}=0\)
\(\textbf{8)}\) Find all 6 trig functions if \(\theta=240^\circ\) \(\sin240^\circ=-\frac{\sqrt{3}}{2}\,\,\,\,\,\,\cos240^\circ=-\frac{1}{2}\,\,\,\,\,\,\tan240^\circ=\sqrt{3}\)
\(\csc240^\circ=-\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\sec240^\circ=-2\,\,\,\,\,\,\cot240^\circ=\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,240^\circ\text{ has reference angle }60^\circ\text{ and is in Quadrant III.}\)
\(\,\,\,\,\,\,\sin240^\circ=-\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\cos240^\circ=-\frac{1}{2}\)
\(\,\,\,\,\,\,\tan240^\circ=\frac{\sin240^\circ}{\cos240^\circ}=\sqrt{3}\)
\(\,\,\,\,\,\,\csc240^\circ=\frac{1}{\sin240^\circ}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\sec240^\circ=\frac{1}{\cos240^\circ}=-2\)
\(\,\,\,\,\,\,\cot240^\circ=\frac{1}{\tan240^\circ}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)
\(\csc240^\circ=-\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\sec240^\circ=-2\,\,\,\,\,\,\cot240^\circ=\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,240^\circ\text{ has reference angle }60^\circ\text{ and is in Quadrant III.}\)
\(\,\,\,\,\,\,\sin240^\circ=-\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\cos240^\circ=-\frac{1}{2}\)
\(\,\,\,\,\,\,\tan240^\circ=\frac{\sin240^\circ}{\cos240^\circ}=\sqrt{3}\)
\(\,\,\,\,\,\,\csc240^\circ=\frac{1}{\sin240^\circ}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\sec240^\circ=\frac{1}{\cos240^\circ}=-2\)
\(\,\,\,\,\,\,\cot240^\circ=\frac{1}{\tan240^\circ}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)
\(\textbf{9)}\) Find all 6 trig functions if \(\theta=120^\circ\) \(\sin120^\circ=\frac{\sqrt{3}}{2}\,\,\,\,\,\,\cos120^\circ=-\frac{1}{2}\,\,\,\,\,\,\tan120^\circ=-\sqrt{3}\)
\(\csc120^\circ=\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\sec120^\circ=-2\,\,\,\,\,\,\cot120^\circ=-\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,120^\circ\text{ has reference angle }60^\circ\text{ and is in Quadrant II.}\)
\(\,\,\,\,\,\,\sin120^\circ=\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\cos120^\circ=-\frac{1}{2}\)
\(\,\,\,\,\,\,\tan120^\circ=\frac{\sin120^\circ}{\cos120^\circ}=-\sqrt{3}\)
\(\,\,\,\,\,\,\csc120^\circ=\frac{1}{\sin120^\circ}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\sec120^\circ=\frac{1}{\cos120^\circ}=-2\)
\(\,\,\,\,\,\,\cot120^\circ=\frac{1}{\tan120^\circ}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}\)
\(\csc120^\circ=\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\sec120^\circ=-2\,\,\,\,\,\,\cot120^\circ=-\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,120^\circ\text{ has reference angle }60^\circ\text{ and is in Quadrant II.}\)
\(\,\,\,\,\,\,\sin120^\circ=\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\cos120^\circ=-\frac{1}{2}\)
\(\,\,\,\,\,\,\tan120^\circ=\frac{\sin120^\circ}{\cos120^\circ}=-\sqrt{3}\)
\(\,\,\,\,\,\,\csc120^\circ=\frac{1}{\sin120^\circ}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\sec120^\circ=\frac{1}{\cos120^\circ}=-2\)
\(\,\,\,\,\,\,\cot120^\circ=\frac{1}{\tan120^\circ}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}\)
\(\textbf{10)}\) Find all 6 trig functions if the terminal side contains the point \((5,-12)\) \(\sin\theta=-\frac{12}{13}\,\,\,\,\,\,\cos\theta=\frac{5}{13}\,\,\,\,\,\,\tan\theta=-\frac{12}{5}\)
\(\csc\theta=-\frac{13}{12}\,\,\,\,\,\,\sec\theta=\frac{13}{5}\,\,\,\,\,\,\cot\theta=-\frac{5}{12}\)
\(\,\,\,\,\,\,x=5,\quad y=-12\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{5^2+(-12)^2}=13\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-\frac{12}{13}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=\frac{5}{13}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=-\frac{12}{5}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-\frac{13}{12}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=\frac{13}{5}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=-\frac{5}{12}\)
\(\csc\theta=-\frac{13}{12}\,\,\,\,\,\,\sec\theta=\frac{13}{5}\,\,\,\,\,\,\cot\theta=-\frac{5}{12}\)
\(\,\,\,\,\,\,x=5,\quad y=-12\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{5^2+(-12)^2}=13\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-\frac{12}{13}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=\frac{5}{13}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=-\frac{12}{5}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-\frac{13}{12}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=\frac{13}{5}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=-\frac{5}{12}\)
\(\textbf{11)}\) Find all 6 trig functions if the terminal side contains the point \((-8,15)\) \(\sin\theta=\frac{15}{17}\,\,\,\,\,\,\cos\theta=-\frac{8}{17}\,\,\,\,\,\,\tan\theta=-\frac{15}{8}\)
\(\csc\theta=\frac{17}{15}\,\,\,\,\,\,\sec\theta=-\frac{17}{8}\,\,\,\,\,\,\cot\theta=-\frac{8}{15}\)
\(\,\,\,\,\,\,x=-8,\quad y=15\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{(-8)^2+15^2}=17\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=\frac{15}{17}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=-\frac{8}{17}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=-\frac{15}{8}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=\frac{17}{15}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=-\frac{17}{8}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=-\frac{8}{15}\)
\(\csc\theta=\frac{17}{15}\,\,\,\,\,\,\sec\theta=-\frac{17}{8}\,\,\,\,\,\,\cot\theta=-\frac{8}{15}\)
\(\,\,\,\,\,\,x=-8,\quad y=15\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{(-8)^2+15^2}=17\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=\frac{15}{17}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=-\frac{8}{17}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=-\frac{15}{8}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=\frac{17}{15}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=-\frac{17}{8}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=-\frac{8}{15}\)
\(\textbf{12)}\) Find all 6 trig functions if \(\cos\theta=\frac{5}{13}\), Quadrant IV \(\sin\theta=-\frac{12}{13}\,\,\,\,\,\,\cos\theta=\frac{5}{13}\,\,\,\,\,\,\tan\theta=-\frac{12}{5}\)
\(\csc\theta=-\frac{13}{12}\,\,\,\,\,\,\sec\theta=\frac{13}{5}\,\,\,\,\,\,\cot\theta=-\frac{5}{12}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=\frac{5}{13}\)
\(\,\,\,\,\,\,x=5,\quad r=13\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,5^2+y^2=13^2\)
\(\,\,\,\,\,\,25+y^2=169\)
\(\,\,\,\,\,\,y^2=144\)
\(\,\,\,\,\,\,y=-12\text{ because Quadrant IV has negative }y.\)
\(\,\,\,\,\,\,\sin\theta=-\frac{12}{13},\quad \cos\theta=\frac{5}{13},\quad \tan\theta=-\frac{12}{5}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{13}{12},\quad \sec\theta=\frac{13}{5},\quad \cot\theta=-\frac{5}{12}\)
\(\csc\theta=-\frac{13}{12}\,\,\,\,\,\,\sec\theta=\frac{13}{5}\,\,\,\,\,\,\cot\theta=-\frac{5}{12}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=\frac{5}{13}\)
\(\,\,\,\,\,\,x=5,\quad r=13\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,5^2+y^2=13^2\)
\(\,\,\,\,\,\,25+y^2=169\)
\(\,\,\,\,\,\,y^2=144\)
\(\,\,\,\,\,\,y=-12\text{ because Quadrant IV has negative }y.\)
\(\,\,\,\,\,\,\sin\theta=-\frac{12}{13},\quad \cos\theta=\frac{5}{13},\quad \tan\theta=-\frac{12}{5}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{13}{12},\quad \sec\theta=\frac{13}{5},\quad \cot\theta=-\frac{5}{12}\)
\(\textbf{13)}\) Find all 6 trig functions if \(\tan\theta=-\frac{4}{3}\), Quadrant II \(\sin\theta=\frac{4}{5}\,\,\,\,\,\,\cos\theta=-\frac{3}{5}\,\,\,\,\,\,\tan\theta=-\frac{4}{3}\)
\(\csc\theta=\frac{5}{4}\,\,\,\,\,\,\sec\theta=-\frac{5}{3}\,\,\,\,\,\,\cot\theta=-\frac{3}{4}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=-\frac{4}{3}\)
\(\,\,\,\,\,\,\text{In Quadrant II, }x\text{ is negative and }y\text{ is positive.}\)
\(\,\,\,\,\,\,x=-3,\quad y=4\)
\(\,\,\,\,\,\,r=\sqrt{(-3)^2+4^2}=5\)
\(\,\,\,\,\,\,\sin\theta=\frac{4}{5}\)
\(\,\,\,\,\,\,\cos\theta=-\frac{3}{5}\)
\(\,\,\,\,\,\,\tan\theta=-\frac{4}{3}\)
\(\,\,\,\,\,\,\csc\theta=\frac{5}{4},\quad \sec\theta=-\frac{5}{3},\quad \cot\theta=-\frac{3}{4}\)
\(\csc\theta=\frac{5}{4}\,\,\,\,\,\,\sec\theta=-\frac{5}{3}\,\,\,\,\,\,\cot\theta=-\frac{3}{4}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=-\frac{4}{3}\)
\(\,\,\,\,\,\,\text{In Quadrant II, }x\text{ is negative and }y\text{ is positive.}\)
\(\,\,\,\,\,\,x=-3,\quad y=4\)
\(\,\,\,\,\,\,r=\sqrt{(-3)^2+4^2}=5\)
\(\,\,\,\,\,\,\sin\theta=\frac{4}{5}\)
\(\,\,\,\,\,\,\cos\theta=-\frac{3}{5}\)
\(\,\,\,\,\,\,\tan\theta=-\frac{4}{3}\)
\(\,\,\,\,\,\,\csc\theta=\frac{5}{4},\quad \sec\theta=-\frac{5}{3},\quad \cot\theta=-\frac{3}{4}\)
\(\textbf{14)}\) Find all 6 trig functions if \(\sec\theta=-\frac{17}{8}\), Quadrant II \(\sin\theta=\frac{15}{17}\,\,\,\,\,\,\cos\theta=-\frac{8}{17}\,\,\,\,\,\,\tan\theta=-\frac{15}{8}\)
\(\csc\theta=\frac{17}{15}\,\,\,\,\,\,\sec\theta=-\frac{17}{8}\,\,\,\,\,\,\cot\theta=-\frac{8}{15}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=-\frac{17}{8}\)
\(\,\,\,\,\,\,r=17,\quad x=-8\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,(-8)^2+y^2=17^2\)
\(\,\,\,\,\,\,64+y^2=289\)
\(\,\,\,\,\,\,y^2=225\)
\(\,\,\,\,\,\,y=15\text{ because Quadrant II has positive }y.\)
\(\,\,\,\,\,\,\sin\theta=\frac{15}{17},\quad \cos\theta=-\frac{8}{17},\quad \tan\theta=-\frac{15}{8}\)
\(\,\,\,\,\,\,\csc\theta=\frac{17}{15},\quad \sec\theta=-\frac{17}{8},\quad \cot\theta=-\frac{8}{15}\)
\(\csc\theta=\frac{17}{15}\,\,\,\,\,\,\sec\theta=-\frac{17}{8}\,\,\,\,\,\,\cot\theta=-\frac{8}{15}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=-\frac{17}{8}\)
\(\,\,\,\,\,\,r=17,\quad x=-8\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,(-8)^2+y^2=17^2\)
\(\,\,\,\,\,\,64+y^2=289\)
\(\,\,\,\,\,\,y^2=225\)
\(\,\,\,\,\,\,y=15\text{ because Quadrant II has positive }y.\)
\(\,\,\,\,\,\,\sin\theta=\frac{15}{17},\quad \cos\theta=-\frac{8}{17},\quad \tan\theta=-\frac{15}{8}\)
\(\,\,\,\,\,\,\csc\theta=\frac{17}{15},\quad \sec\theta=-\frac{17}{8},\quad \cot\theta=-\frac{8}{15}\)
\(\textbf{15)}\) Find all 6 trig functions if \(\csc\theta=-\frac{25}{7}\), Quadrant III \(\sin\theta=-\frac{7}{25}\,\,\,\,\,\,\cos\theta=-\frac{24}{25}\,\,\,\,\,\,\tan\theta=\frac{7}{24}\)
\(\csc\theta=-\frac{25}{7}\,\,\,\,\,\,\sec\theta=-\frac{25}{24}\,\,\,\,\,\,\cot\theta=\frac{24}{7}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-\frac{25}{7}\)
\(\,\,\,\,\,\,r=25,\quad y=-7\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,x^2+(-7)^2=25^2\)
\(\,\,\,\,\,\,x^2+49=625\)
\(\,\,\,\,\,\,x^2=576\)
\(\,\,\,\,\,\,x=-24\text{ because Quadrant III has negative }x.\)
\(\,\,\,\,\,\,\sin\theta=-\frac{7}{25},\quad \cos\theta=-\frac{24}{25},\quad \tan\theta=\frac{7}{24}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{25}{7},\quad \sec\theta=-\frac{25}{24},\quad \cot\theta=\frac{24}{7}\)
\(\csc\theta=-\frac{25}{7}\,\,\,\,\,\,\sec\theta=-\frac{25}{24}\,\,\,\,\,\,\cot\theta=\frac{24}{7}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-\frac{25}{7}\)
\(\,\,\,\,\,\,r=25,\quad y=-7\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,x^2+(-7)^2=25^2\)
\(\,\,\,\,\,\,x^2+49=625\)
\(\,\,\,\,\,\,x^2=576\)
\(\,\,\,\,\,\,x=-24\text{ because Quadrant III has negative }x.\)
\(\,\,\,\,\,\,\sin\theta=-\frac{7}{25},\quad \cos\theta=-\frac{24}{25},\quad \tan\theta=\frac{7}{24}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{25}{7},\quad \sec\theta=-\frac{25}{24},\quad \cot\theta=\frac{24}{7}\)
\(\textbf{16)}\) Find all 6 trig functions if \(\sin\theta=\frac{3}{5}\) and \(\tan\theta\lt0\) \(\sin\theta=\frac{3}{5}\,\,\,\,\,\,\cos\theta=-\frac{4}{5}\,\,\,\,\,\,\tan\theta=-\frac{3}{4}\)
\(\csc\theta=\frac{5}{3}\,\,\,\,\,\,\sec\theta=-\frac{5}{4}\,\,\,\,\,\,\cot\theta=-\frac{4}{3}\)
\(\,\,\,\,\,\,\sin\theta=\frac{3}{5}\text{ means }y=3\text{ and }r=5.\)
\(\,\,\,\,\,\,\tan\theta\lt0\text{ and }\sin\theta\gt0\text{ means Quadrant II.}\)
\(\,\,\,\,\,\,x^2+3^2=5^2\)
\(\,\,\,\,\,\,x^2=16\)
\(\,\,\,\,\,\,x=-4\)
\(\,\,\,\,\,\,\sin\theta=\frac{3}{5},\quad \cos\theta=-\frac{4}{5},\quad \tan\theta=-\frac{3}{4}\)
\(\,\,\,\,\,\,\csc\theta=\frac{5}{3},\quad \sec\theta=-\frac{5}{4},\quad \cot\theta=-\frac{4}{3}\)
\(\csc\theta=\frac{5}{3}\,\,\,\,\,\,\sec\theta=-\frac{5}{4}\,\,\,\,\,\,\cot\theta=-\frac{4}{3}\)
\(\,\,\,\,\,\,\sin\theta=\frac{3}{5}\text{ means }y=3\text{ and }r=5.\)
\(\,\,\,\,\,\,\tan\theta\lt0\text{ and }\sin\theta\gt0\text{ means Quadrant II.}\)
\(\,\,\,\,\,\,x^2+3^2=5^2\)
\(\,\,\,\,\,\,x^2=16\)
\(\,\,\,\,\,\,x=-4\)
\(\,\,\,\,\,\,\sin\theta=\frac{3}{5},\quad \cos\theta=-\frac{4}{5},\quad \tan\theta=-\frac{3}{4}\)
\(\,\,\,\,\,\,\csc\theta=\frac{5}{3},\quad \sec\theta=-\frac{5}{4},\quad \cot\theta=-\frac{4}{3}\)
\(\textbf{17)}\) Find all 6 trig functions if \(\cos\theta=-\frac{12}{13}\) and \(\sin\theta\lt0\) \(\sin\theta=-\frac{5}{13}\,\,\,\,\,\,\cos\theta=-\frac{12}{13}\,\,\,\,\,\,\tan\theta=\frac{5}{12}\)
\(\csc\theta=-\frac{13}{5}\,\,\,\,\,\,\sec\theta=-\frac{13}{12}\,\,\,\,\,\,\cot\theta=\frac{12}{5}\)
\(\,\,\,\,\,\,\cos\theta=-\frac{12}{13}\text{ means }x=-12\text{ and }r=13.\)
\(\,\,\,\,\,\,\sin\theta\lt0\text{ means }y\text{ is negative, so this is Quadrant III.}\)
\(\,\,\,\,\,\,(-12)^2+y^2=13^2\)
\(\,\,\,\,\,\,144+y^2=169\)
\(\,\,\,\,\,\,y^2=25\)
\(\,\,\,\,\,\,y=-5\)
\(\,\,\,\,\,\,\sin\theta=-\frac{5}{13},\quad \cos\theta=-\frac{12}{13},\quad \tan\theta=\frac{5}{12}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{13}{5},\quad \sec\theta=-\frac{13}{12},\quad \cot\theta=\frac{12}{5}\)
\(\csc\theta=-\frac{13}{5}\,\,\,\,\,\,\sec\theta=-\frac{13}{12}\,\,\,\,\,\,\cot\theta=\frac{12}{5}\)
\(\,\,\,\,\,\,\cos\theta=-\frac{12}{13}\text{ means }x=-12\text{ and }r=13.\)
\(\,\,\,\,\,\,\sin\theta\lt0\text{ means }y\text{ is negative, so this is Quadrant III.}\)
\(\,\,\,\,\,\,(-12)^2+y^2=13^2\)
\(\,\,\,\,\,\,144+y^2=169\)
\(\,\,\,\,\,\,y^2=25\)
\(\,\,\,\,\,\,y=-5\)
\(\,\,\,\,\,\,\sin\theta=-\frac{5}{13},\quad \cos\theta=-\frac{12}{13},\quad \tan\theta=\frac{5}{12}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{13}{5},\quad \sec\theta=-\frac{13}{12},\quad \cot\theta=\frac{12}{5}\)
\(\textbf{18)}\) Find all 6 trig functions if the terminal side contains the point \((0,-6)\) \(\sin\theta=-1\,\,\,\,\,\,\cos\theta=0\,\,\,\,\,\,\tan\theta=\text{undefined}\)
\(\csc\theta=-1\,\,\,\,\,\,\sec\theta=\text{undefined}\,\,\,\,\,\,\cot\theta=0\)
\(\,\,\,\,\,\,x=0,\quad y=-6\)
\(\,\,\,\,\,\,r=\sqrt{0^2+(-6)^2}=6\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-1\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=0\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{-6}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-1\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=\frac{6}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{0}{-6}=0\)
\(\csc\theta=-1\,\,\,\,\,\,\sec\theta=\text{undefined}\,\,\,\,\,\,\cot\theta=0\)
\(\,\,\,\,\,\,x=0,\quad y=-6\)
\(\,\,\,\,\,\,r=\sqrt{0^2+(-6)^2}=6\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-1\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=0\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{-6}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-1\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=\frac{6}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{0}{-6}=0\)
\(\textbf{19)}\) Find all 6 trig functions if \(\theta=\frac{5\pi}{6}\) \(\sin\frac{5\pi}{6}=\frac{1}{2}\,\,\,\,\,\,\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}\,\,\,\,\,\,\tan\frac{5\pi}{6}=-\frac{\sqrt{3}}{3}\)
\(\csc\frac{5\pi}{6}=2\,\,\,\,\,\,\sec\frac{5\pi}{6}=-\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\cot\frac{5\pi}{6}=-\sqrt{3}\)
\(\,\,\,\,\,\,\frac{5\pi}{6}=150^\circ\)
\(\,\,\,\,\,\,150^\circ\text{ has reference angle }30^\circ\text{ and is in Quadrant II.}\)
\(\,\,\,\,\,\,\sin\frac{5\pi}{6}=\frac{1}{2}\)
\(\,\,\,\,\,\,\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\tan\frac{5\pi}{6}=-\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\csc\frac{5\pi}{6}=2,\quad \sec\frac{5\pi}{6}=-\frac{2\sqrt{3}}{3},\quad \cot\frac{5\pi}{6}=-\sqrt{3}\)
\(\csc\frac{5\pi}{6}=2\,\,\,\,\,\,\sec\frac{5\pi}{6}=-\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\cot\frac{5\pi}{6}=-\sqrt{3}\)
\(\,\,\,\,\,\,\frac{5\pi}{6}=150^\circ\)
\(\,\,\,\,\,\,150^\circ\text{ has reference angle }30^\circ\text{ and is in Quadrant II.}\)
\(\,\,\,\,\,\,\sin\frac{5\pi}{6}=\frac{1}{2}\)
\(\,\,\,\,\,\,\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\tan\frac{5\pi}{6}=-\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\csc\frac{5\pi}{6}=2,\quad \sec\frac{5\pi}{6}=-\frac{2\sqrt{3}}{3},\quad \cot\frac{5\pi}{6}=-\sqrt{3}\)
\(\textbf{20)}\) Find all 6 trig functions if \(\theta=\frac{7\pi}{4}\) \(\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}\,\,\,\,\,\,\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}\,\,\,\,\,\,\tan\frac{7\pi}{4}=-1\)
\(\csc\frac{7\pi}{4}=-\sqrt{2}\,\,\,\,\,\,\sec\frac{7\pi}{4}=\sqrt{2}\,\,\,\,\,\,\cot\frac{7\pi}{4}=-1\)
\(\,\,\,\,\,\,\frac{7\pi}{4}=315^\circ\)
\(\,\,\,\,\,\,315^\circ\text{ has reference angle }45^\circ\text{ and is in Quadrant IV.}\)
\(\,\,\,\,\,\,\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\tan\frac{7\pi}{4}=-1\)
\(\,\,\,\,\,\,\csc\frac{7\pi}{4}=-\sqrt{2},\quad \sec\frac{7\pi}{4}=\sqrt{2},\quad \cot\frac{7\pi}{4}=-1\)
\(\csc\frac{7\pi}{4}=-\sqrt{2}\,\,\,\,\,\,\sec\frac{7\pi}{4}=\sqrt{2}\,\,\,\,\,\,\cot\frac{7\pi}{4}=-1\)
\(\,\,\,\,\,\,\frac{7\pi}{4}=315^\circ\)
\(\,\,\,\,\,\,315^\circ\text{ has reference angle }45^\circ\text{ and is in Quadrant IV.}\)
\(\,\,\,\,\,\,\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\tan\frac{7\pi}{4}=-1\)
\(\,\,\,\,\,\,\csc\frac{7\pi}{4}=-\sqrt{2},\quad \sec\frac{7\pi}{4}=\sqrt{2},\quad \cot\frac{7\pi}{4}=-1\)
See Related Pages\(\)
\(\bullet\text{ Right Triangle Trigonometry}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\)
\(\bullet\text{ Angle of Depression and Elevation}\)
\(\,\,\,\,\,\,\,\,\text{Angle of Depression}=\text{Angle of Elevation}…\)
\(\bullet\text{ Convert to Radians and to Degrees}\)
\(\,\,\,\,\,\,\,\,\text{Radians} \rightarrow \text{Degrees}, \times \displaystyle \frac{180^{\circ}}{\pi}…\)
\(\bullet\text{ Degrees, Minutes and Seconds}\)
\(\,\,\,\,\,\,\,\,48^{\circ}34’21”…\)
\(\bullet\text{ Coterminal Angles}\)
\(\,\,\,\,\,\,\,\,\pm 360^{\circ} \text { or } \pm 2\pi n…\)
\(\bullet\text{ Reference Angles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Find All 6 Trig Functions}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Unit Circle}\)
\(\,\,\,\,\,\,\,\,\sin{(60^{\circ})}=\displaystyle\frac{\sqrt{3}}{2}…\)
\(\bullet\text{ Law of Sines}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\sin{A}}{a}=\frac{\sin{B}}{b}=\frac{\sin{C}}{c}\) 
\(…\)
\(\bullet\text{ Area of SAS Triangles}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\frac{1}{2}ab \sin{C}\) \(…\)
\(\bullet\text{ Law of Cosines}\)
\(\,\,\,\,\,\,\,\,a^2=b^2+c^2-2bc \cos{A}\) \(…\)
\(\bullet\text{ Area of SSS Triangles (Heron’s formula)}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\sqrt{s(s-a)(s-b)(s-c)}\) \(…\)
\(\bullet\text{ Geometric Mean}\)
\(\,\,\,\,\,\,\,\,x=\sqrt{ab} \text{ or } \displaystyle\frac{a}{x}=\frac{x}{b}…\)
\(\bullet\text{ Geometric Mean- Similar Right Triangles}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Inverse Trigonmetric Functions}\)
\(\,\,\,\,\,\,\,\,\sin {\left(cos^{-1}\left(\frac{3}{5}\right)\right)}…\)
\(\bullet\text{ Sum and Difference of Angles Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(A+B)}=\sin{A}\cos{B}+\cos{A}\sin{B}…\)
\(\bullet\text{ Double-Angle and Half-Angle Formulas}\)
\(\,\,\,\,\,\,\,\,\sin{(2A)}=2\sin{(A)}\cos{(A)}…\)
\(\bullet\text{ Trigonometry-Pythagorean Identities}\)
\(\,\,\,\,\,\,\,\,\sin^2{(x)}+\cos^2{(x)}=1…\)
\(\bullet\text{ Product-Sum Identities}\)
\(\,\,\,\,\,\,\,\,\cos{\alpha}\cos{\beta}=\left(\displaystyle\frac{\cos{(\alpha+\beta)}+\cos{(\alpha-\beta)}}{2}\right)…\)
\(\bullet\text{ Cofunction Identities}\)
\(\,\,\,\,\,\,\,\,\sin{(x)}=\cos{(\frac{\pi}{2}-x)}…\)
\(\bullet\text{ Proving Trigonometric Identities}\)
\(\,\,\,\,\,\,\,\,\sec{x}-\cos{x}=\displaystyle\frac{\tan^2{x}}{\sec{x}}…\)
\(\bullet\text{ Graphing Trig Functions- sin and cos}\)
\(\,\,\,\,\,\,\,\,f(x)=A \sin{B(x-c)}+D \) 
\(…\)
\(\bullet\text{ Solving Trigonometric Equations}\)
\(\,\,\,\,\,\,\,\,2\cos{(x)}=\sqrt{3}…\)