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 \(\theta=180^{\circ}\)
\(\sin180^{\circ}=0\,\,\,\,\,\,\cos180^{\circ}=-1\,\,\,\,\,\,\tan180^{\circ}=0\)
\(\csc180^{\circ}=\text{undefined}\,\,\,\,\,\,\sec180^{\circ}=-1\,\,\,\,\,\,\cot180^{\circ}=\text{undefined}\)
\(\,\,\,\,\,\,180^\circ\text{ is on the negative }x\text{-axis.}\)
\(\,\,\,\,\,\,\left(x,y\right)=\left(-1,0\right)\)
\(\,\,\,\,\,\,\sin\theta=y=0\)
\(\,\,\,\,\,\,\cos\theta=x=-1\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{0}{-1}=0\)
\(\,\,\,\,\,\,\csc\theta=\frac{1}{\sin\theta}=\frac{1}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\sec\theta=\frac{1}{\cos\theta}=\frac{1}{-1}=-1\)
\(\,\,\,\,\,\,\cot\theta=\frac{1}{\tan\theta}=\frac{1}{0}=\text{undefined}\)
\(\textbf{2)}\) Find all 6 trig functions if \(\theta=330^{\circ}\)
\(\sin330^{\circ}=-\frac{1}{2}\,\,\,\,\,\,\cos330^{\circ}=\frac{\sqrt{3}}{2}\,\,\,\,\,\,\tan330^{\circ}=-\frac{\sqrt{3}}{3}\)
\(\csc330^{\circ}=-2\,\,\,\,\,\,\sec330^{\circ}=\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\cot330^{\circ}=-\sqrt{3}\)
\(\,\,\,\,\,\,330^\circ\text{ has reference angle }30^\circ\text{ and is in Quadrant IV.}\)
\(\,\,\,\,\,\,\sin330^\circ=-\frac{1}{2}\)
\(\,\,\,\,\,\,\cos330^\circ=\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\tan330^\circ=\frac{\sin330^\circ}{\cos330^\circ}=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\csc330^\circ=\frac{1}{\sin330^\circ}=-2\)
\(\,\,\,\,\,\,\sec330^\circ=\frac{1}{\cos330^\circ}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\cot330^\circ=\frac{1}{\tan330^\circ}=-\sqrt{3}\)
\(\textbf{3)}\) Find all 6 trig functions if \(\theta=-135^{\circ}\)
\(\sin(-135^{\circ})=-\frac{\sqrt{2}}{2}\,\,\,\,\,\,\cos(-135^{\circ})=-\frac{\sqrt{2}}{2}\,\,\,\,\,\,\tan(-135^{\circ})=1\)
\(\csc(-135^{\circ})=-\sqrt{2}\,\,\,\,\,\,\sec(-135^{\circ})=-\sqrt{2}\,\,\,\,\,\,\cot(-135^{\circ})=1\)
\(\,\,\,\,\,\,-135^\circ\text{ is coterminal with }225^\circ.\)
\(\,\,\,\,\,\,225^\circ\text{ has reference angle }45^\circ\text{ and is in Quadrant III.}\)
\(\,\,\,\,\,\,\sin\theta=-\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\cos\theta=-\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\tan\theta=\frac{\sin\theta}{\cos\theta}=1\)
\(\,\,\,\,\,\,\csc\theta=\frac{1}{\sin\theta}=-\sqrt{2}\)
\(\,\,\,\,\,\,\sec\theta=\frac{1}{\cos\theta}=-\sqrt{2}\)
\(\,\,\,\,\,\,\cot\theta=\frac{1}{\tan\theta}=1\)
\(\textbf{4)}\) Find all 6 trig functions if the terminal side contains the point \((3,4)\)
\(\sin\theta=\frac{4}{5}\,\,\,\,\,\,\cos\theta=\frac{3}{5}\,\,\,\,\,\,\tan\theta=\frac{4}{3}\)
\(\csc\theta=\frac{5}{4}\,\,\,\,\,\,\sec\theta=\frac{5}{3}\,\,\,\,\,\,\cot\theta=\frac{3}{4}\)
\(\,\,\,\,\,\,x=3,\quad y=4\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{3^2+4^2}=5\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=\frac{4}{5}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=\frac{3}{5}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{4}{3}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=\frac{5}{4}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=\frac{5}{3}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{3}{4}\)
\(\textbf{5)}\) Find all 6 trig functions if the terminal side contains the point \((-5,-12)\)
\(\sin\theta=-\frac{12}{13}\,\,\,\,\,\,\cos\theta=-\frac{5}{13}\,\,\,\,\,\,\tan\theta=\frac{12}{5}\)
\(\csc\theta=-\frac{13}{12}\,\,\,\,\,\,\sec\theta=-\frac{13}{5}\,\,\,\,\,\,\cot\theta=\frac{5}{12}\)
\(\,\,\,\,\,\,x=-5,\quad y=-12\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{(-5)^2+(-12)^2}=13\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-\frac{12}{13}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=-\frac{5}{13}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{-12}{-5}=\frac{12}{5}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-\frac{13}{12}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=-\frac{13}{5}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{-5}{-12}=\frac{5}{12}\)
\(\textbf{6)}\) Find all 6 trig functions if \(\sin(\theta)=-\frac{4}{5}\), Quadrant III
\(\sin\theta=-\frac{4}{5}\,\,\,\,\,\,\cos\theta=-\frac{3}{5}\,\,\,\,\,\,\tan\theta=\frac{4}{3}\)
\(\csc\theta=-\frac{5}{4}\,\,\,\,\,\,\sec\theta=-\frac{5}{3}\,\,\,\,\,\,\cot\theta=\frac{3}{4}\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-\frac{4}{5}\)
\(\,\,\,\,\,\,y=-4,\quad r=5\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,x^2+(-4)^2=5^2\)
\(\,\,\,\,\,\,x^2+16=25\)
\(\,\,\,\,\,\,x^2=9\)
\(\,\,\,\,\,\,x=-3\text{ because Quadrant III has negative }x\text{ and negative }y.\)
\(\,\,\,\,\,\,\sin\theta=-\frac{4}{5},\quad \cos\theta=-\frac{3}{5},\quad \tan\theta=\frac{4}{3}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{5}{4},\quad \sec\theta=-\frac{5}{3},\quad \cot\theta=\frac{3}{4}\)
\(\textbf{7)}\) Find all 6 trig functions if \(\theta=90^\circ\)
\(\sin90^\circ=1\,\,\,\,\,\,\cos90^\circ=0\,\,\,\,\,\,\tan90^\circ=\text{undefined}\)
\(\csc90^\circ=1\,\,\,\,\,\,\sec90^\circ=\text{undefined}\,\,\,\,\,\,\cot90^\circ=0\)
\(\,\,\,\,\,\,90^\circ\text{ is on the positive }y\text{-axis.}\)
\(\,\,\,\,\,\,\left(x,y\right)=\left(0,1\right)\)
\(\,\,\,\,\,\,\sin\theta=y=1\)
\(\,\,\,\,\,\,\cos\theta=x=0\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{1}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\csc\theta=\frac{1}{\sin\theta}=1\)
\(\,\,\,\,\,\,\sec\theta=\frac{1}{\cos\theta}=\frac{1}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{0}{1}=0\)
\(\textbf{8)}\) Find all 6 trig functions if \(\theta=240^\circ\)
\(\sin240^\circ=-\frac{\sqrt{3}}{2}\,\,\,\,\,\,\cos240^\circ=-\frac{1}{2}\,\,\,\,\,\,\tan240^\circ=\sqrt{3}\)
\(\csc240^\circ=-\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\sec240^\circ=-2\,\,\,\,\,\,\cot240^\circ=\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,240^\circ\text{ has reference angle }60^\circ\text{ and is in Quadrant III.}\)
\(\,\,\,\,\,\,\sin240^\circ=-\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\cos240^\circ=-\frac{1}{2}\)
\(\,\,\,\,\,\,\tan240^\circ=\frac{\sin240^\circ}{\cos240^\circ}=\sqrt{3}\)
\(\,\,\,\,\,\,\csc240^\circ=\frac{1}{\sin240^\circ}=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\sec240^\circ=\frac{1}{\cos240^\circ}=-2\)
\(\,\,\,\,\,\,\cot240^\circ=\frac{1}{\tan240^\circ}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}\)
\(\textbf{9)}\) Find all 6 trig functions if \(\theta=120^\circ\)
\(\sin120^\circ=\frac{\sqrt{3}}{2}\,\,\,\,\,\,\cos120^\circ=-\frac{1}{2}\,\,\,\,\,\,\tan120^\circ=-\sqrt{3}\)
\(\csc120^\circ=\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\sec120^\circ=-2\,\,\,\,\,\,\cot120^\circ=-\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,120^\circ\text{ has reference angle }60^\circ\text{ and is in Quadrant II.}\)
\(\,\,\,\,\,\,\sin120^\circ=\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\cos120^\circ=-\frac{1}{2}\)
\(\,\,\,\,\,\,\tan120^\circ=\frac{\sin120^\circ}{\cos120^\circ}=-\sqrt{3}\)
\(\,\,\,\,\,\,\csc120^\circ=\frac{1}{\sin120^\circ}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\sec120^\circ=\frac{1}{\cos120^\circ}=-2\)
\(\,\,\,\,\,\,\cot120^\circ=\frac{1}{\tan120^\circ}=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}\)
\(\textbf{10)}\) Find all 6 trig functions if the terminal side contains the point \((5,-12)\)
\(\sin\theta=-\frac{12}{13}\,\,\,\,\,\,\cos\theta=\frac{5}{13}\,\,\,\,\,\,\tan\theta=-\frac{12}{5}\)
\(\csc\theta=-\frac{13}{12}\,\,\,\,\,\,\sec\theta=\frac{13}{5}\,\,\,\,\,\,\cot\theta=-\frac{5}{12}\)
\(\,\,\,\,\,\,x=5,\quad y=-12\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{5^2+(-12)^2}=13\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-\frac{12}{13}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=\frac{5}{13}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=-\frac{12}{5}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-\frac{13}{12}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=\frac{13}{5}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=-\frac{5}{12}\)
\(\textbf{11)}\) Find all 6 trig functions if the terminal side contains the point \((-8,15)\)
\(\sin\theta=\frac{15}{17}\,\,\,\,\,\,\cos\theta=-\frac{8}{17}\,\,\,\,\,\,\tan\theta=-\frac{15}{8}\)
\(\csc\theta=\frac{17}{15}\,\,\,\,\,\,\sec\theta=-\frac{17}{8}\,\,\,\,\,\,\cot\theta=-\frac{8}{15}\)
\(\,\,\,\,\,\,x=-8,\quad y=15\)
\(\,\,\,\,\,\,r=\sqrt{x^2+y^2}=\sqrt{(-8)^2+15^2}=17\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=\frac{15}{17}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=-\frac{8}{17}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=-\frac{15}{8}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=\frac{17}{15}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=-\frac{17}{8}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=-\frac{8}{15}\)
\(\textbf{12)}\) Find all 6 trig functions if \(\cos\theta=\frac{5}{13}\), Quadrant IV
\(\sin\theta=-\frac{12}{13}\,\,\,\,\,\,\cos\theta=\frac{5}{13}\,\,\,\,\,\,\tan\theta=-\frac{12}{5}\)
\(\csc\theta=-\frac{13}{12}\,\,\,\,\,\,\sec\theta=\frac{13}{5}\,\,\,\,\,\,\cot\theta=-\frac{5}{12}\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=\frac{5}{13}\)
\(\,\,\,\,\,\,x=5,\quad r=13\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,5^2+y^2=13^2\)
\(\,\,\,\,\,\,25+y^2=169\)
\(\,\,\,\,\,\,y^2=144\)
\(\,\,\,\,\,\,y=-12\text{ because Quadrant IV has negative }y.\)
\(\,\,\,\,\,\,\sin\theta=-\frac{12}{13},\quad \cos\theta=\frac{5}{13},\quad \tan\theta=-\frac{12}{5}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{13}{12},\quad \sec\theta=\frac{13}{5},\quad \cot\theta=-\frac{5}{12}\)
\(\textbf{13)}\) Find all 6 trig functions if \(\tan\theta=-\frac{4}{3}\), Quadrant II
\(\sin\theta=\frac{4}{5}\,\,\,\,\,\,\cos\theta=-\frac{3}{5}\,\,\,\,\,\,\tan\theta=-\frac{4}{3}\)
\(\csc\theta=\frac{5}{4}\,\,\,\,\,\,\sec\theta=-\frac{5}{3}\,\,\,\,\,\,\cot\theta=-\frac{3}{4}\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=-\frac{4}{3}\)
\(\,\,\,\,\,\,\text{In Quadrant II, }x\text{ is negative and }y\text{ is positive.}\)
\(\,\,\,\,\,\,x=-3,\quad y=4\)
\(\,\,\,\,\,\,r=\sqrt{(-3)^2+4^2}=5\)
\(\,\,\,\,\,\,\sin\theta=\frac{4}{5}\)
\(\,\,\,\,\,\,\cos\theta=-\frac{3}{5}\)
\(\,\,\,\,\,\,\tan\theta=-\frac{4}{3}\)
\(\,\,\,\,\,\,\csc\theta=\frac{5}{4},\quad \sec\theta=-\frac{5}{3},\quad \cot\theta=-\frac{3}{4}\)
\(\textbf{14)}\) Find all 6 trig functions if \(\sec\theta=-\frac{17}{8}\), Quadrant II
\(\sin\theta=\frac{15}{17}\,\,\,\,\,\,\cos\theta=-\frac{8}{17}\,\,\,\,\,\,\tan\theta=-\frac{15}{8}\)
\(\csc\theta=\frac{17}{15}\,\,\,\,\,\,\sec\theta=-\frac{17}{8}\,\,\,\,\,\,\cot\theta=-\frac{8}{15}\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=-\frac{17}{8}\)
\(\,\,\,\,\,\,r=17,\quad x=-8\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,(-8)^2+y^2=17^2\)
\(\,\,\,\,\,\,64+y^2=289\)
\(\,\,\,\,\,\,y^2=225\)
\(\,\,\,\,\,\,y=15\text{ because Quadrant II has positive }y.\)
\(\,\,\,\,\,\,\sin\theta=\frac{15}{17},\quad \cos\theta=-\frac{8}{17},\quad \tan\theta=-\frac{15}{8}\)
\(\,\,\,\,\,\,\csc\theta=\frac{17}{15},\quad \sec\theta=-\frac{17}{8},\quad \cot\theta=-\frac{8}{15}\)
\(\textbf{15)}\) Find all 6 trig functions if \(\csc\theta=-\frac{25}{7}\), Quadrant III
\(\sin\theta=-\frac{7}{25}\,\,\,\,\,\,\cos\theta=-\frac{24}{25}\,\,\,\,\,\,\tan\theta=\frac{7}{24}\)
\(\csc\theta=-\frac{25}{7}\,\,\,\,\,\,\sec\theta=-\frac{25}{24}\,\,\,\,\,\,\cot\theta=\frac{24}{7}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-\frac{25}{7}\)
\(\,\,\,\,\,\,r=25,\quad y=-7\)
\(\,\,\,\,\,\,x^2+y^2=r^2\)
\(\,\,\,\,\,\,x^2+(-7)^2=25^2\)
\(\,\,\,\,\,\,x^2+49=625\)
\(\,\,\,\,\,\,x^2=576\)
\(\,\,\,\,\,\,x=-24\text{ because Quadrant III has negative }x.\)
\(\,\,\,\,\,\,\sin\theta=-\frac{7}{25},\quad \cos\theta=-\frac{24}{25},\quad \tan\theta=\frac{7}{24}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{25}{7},\quad \sec\theta=-\frac{25}{24},\quad \cot\theta=\frac{24}{7}\)
\(\textbf{16)}\) Find all 6 trig functions if \(\sin\theta=\frac{3}{5}\) and \(\tan\theta\lt0\)
\(\sin\theta=\frac{3}{5}\,\,\,\,\,\,\cos\theta=-\frac{4}{5}\,\,\,\,\,\,\tan\theta=-\frac{3}{4}\)
\(\csc\theta=\frac{5}{3}\,\,\,\,\,\,\sec\theta=-\frac{5}{4}\,\,\,\,\,\,\cot\theta=-\frac{4}{3}\)
\(\,\,\,\,\,\,\sin\theta=\frac{3}{5}\text{ means }y=3\text{ and }r=5.\)
\(\,\,\,\,\,\,\tan\theta\lt0\text{ and }\sin\theta\gt0\text{ means Quadrant II.}\)
\(\,\,\,\,\,\,x^2+3^2=5^2\)
\(\,\,\,\,\,\,x^2=16\)
\(\,\,\,\,\,\,x=-4\)
\(\,\,\,\,\,\,\sin\theta=\frac{3}{5},\quad \cos\theta=-\frac{4}{5},\quad \tan\theta=-\frac{3}{4}\)
\(\,\,\,\,\,\,\csc\theta=\frac{5}{3},\quad \sec\theta=-\frac{5}{4},\quad \cot\theta=-\frac{4}{3}\)
\(\textbf{17)}\) Find all 6 trig functions if \(\cos\theta=-\frac{12}{13}\) and \(\sin\theta\lt0\)
\(\sin\theta=-\frac{5}{13}\,\,\,\,\,\,\cos\theta=-\frac{12}{13}\,\,\,\,\,\,\tan\theta=\frac{5}{12}\)
\(\csc\theta=-\frac{13}{5}\,\,\,\,\,\,\sec\theta=-\frac{13}{12}\,\,\,\,\,\,\cot\theta=\frac{12}{5}\)
\(\,\,\,\,\,\,\cos\theta=-\frac{12}{13}\text{ means }x=-12\text{ and }r=13.\)
\(\,\,\,\,\,\,\sin\theta\lt0\text{ means }y\text{ is negative, so this is Quadrant III.}\)
\(\,\,\,\,\,\,(-12)^2+y^2=13^2\)
\(\,\,\,\,\,\,144+y^2=169\)
\(\,\,\,\,\,\,y^2=25\)
\(\,\,\,\,\,\,y=-5\)
\(\,\,\,\,\,\,\sin\theta=-\frac{5}{13},\quad \cos\theta=-\frac{12}{13},\quad \tan\theta=\frac{5}{12}\)
\(\,\,\,\,\,\,\csc\theta=-\frac{13}{5},\quad \sec\theta=-\frac{13}{12},\quad \cot\theta=\frac{12}{5}\)
\(\textbf{18)}\) Find all 6 trig functions if the terminal side contains the point \((0,-6)\)
\(\sin\theta=-1\,\,\,\,\,\,\cos\theta=0\,\,\,\,\,\,\tan\theta=\text{undefined}\)
\(\csc\theta=-1\,\,\,\,\,\,\sec\theta=\text{undefined}\,\,\,\,\,\,\cot\theta=0\)
\(\,\,\,\,\,\,x=0,\quad y=-6\)
\(\,\,\,\,\,\,r=\sqrt{0^2+(-6)^2}=6\)
\(\,\,\,\,\,\,\sin\theta=\frac{y}{r}=-1\)
\(\,\,\,\,\,\,\cos\theta=\frac{x}{r}=0\)
\(\,\,\,\,\,\,\tan\theta=\frac{y}{x}=\frac{-6}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\csc\theta=\frac{r}{y}=-1\)
\(\,\,\,\,\,\,\sec\theta=\frac{r}{x}=\frac{6}{0}=\text{undefined}\)
\(\,\,\,\,\,\,\cot\theta=\frac{x}{y}=\frac{0}{-6}=0\)
\(\textbf{19)}\) Find all 6 trig functions if \(\theta=\frac{5\pi}{6}\)
\(\sin\frac{5\pi}{6}=\frac{1}{2}\,\,\,\,\,\,\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}\,\,\,\,\,\,\tan\frac{5\pi}{6}=-\frac{\sqrt{3}}{3}\)
\(\csc\frac{5\pi}{6}=2\,\,\,\,\,\,\sec\frac{5\pi}{6}=-\frac{2\sqrt{3}}{3}\,\,\,\,\,\,\cot\frac{5\pi}{6}=-\sqrt{3}\)
\(\,\,\,\,\,\,\frac{5\pi}{6}=150^\circ\)
\(\,\,\,\,\,\,150^\circ\text{ has reference angle }30^\circ\text{ and is in Quadrant II.}\)
\(\,\,\,\,\,\,\sin\frac{5\pi}{6}=\frac{1}{2}\)
\(\,\,\,\,\,\,\cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}\)
\(\,\,\,\,\,\,\tan\frac{5\pi}{6}=-\frac{\sqrt{3}}{3}\)
\(\,\,\,\,\,\,\csc\frac{5\pi}{6}=2,\quad \sec\frac{5\pi}{6}=-\frac{2\sqrt{3}}{3},\quad \cot\frac{5\pi}{6}=-\sqrt{3}\)
\(\textbf{20)}\) Find all 6 trig functions if \(\theta=\frac{7\pi}{4}\)
\(\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}\,\,\,\,\,\,\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}\,\,\,\,\,\,\tan\frac{7\pi}{4}=-1\)
\(\csc\frac{7\pi}{4}=-\sqrt{2}\,\,\,\,\,\,\sec\frac{7\pi}{4}=\sqrt{2}\,\,\,\,\,\,\cot\frac{7\pi}{4}=-1\)
\(\,\,\,\,\,\,\frac{7\pi}{4}=315^\circ\)
\(\,\,\,\,\,\,315^\circ\text{ has reference angle }45^\circ\text{ and is in Quadrant IV.}\)
\(\,\,\,\,\,\,\sin\frac{7\pi}{4}=-\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}\)
\(\,\,\,\,\,\,\tan\frac{7\pi}{4}=-1\)
\(\,\,\,\,\,\,\csc\frac{7\pi}{4}=-\sqrt{2},\quad \sec\frac{7\pi}{4}=\sqrt{2},\quad \cot\frac{7\pi}{4}=-1\)
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