## Precalculus (6th Edition) Blitzer

The six trigonometric functions of $\theta$ for point $\left( 5,-5 \right)$ are, $\sin \theta =-\frac{\sqrt{2}}{2},\cos \theta =\frac{\sqrt{2}}{2},\tan \theta =-1,\csc \theta =-\sqrt{2},\sec \theta =\sqrt{2}$ and $\cot \theta =-1$.
Consider the point $\left( 5,-5 \right)$. Here, $x=5$ and $y=-5$. The six trigonometric functions of $\theta$ are defined in terms of a ratio. According to the Pythagoras theorem, the hypotenuse is, $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$ Substitute $5$ for $x$ and $-5$ for $y$. \begin{align} & r=\sqrt{{{\left( 5 \right)}^{2}}+{{\left( -5 \right)}^{2}}} \\ & =\sqrt{25+25} \\ & =\sqrt{50} \\ & =5\sqrt{2} \end{align} Recall the trigonometric expression of $\sin \theta$. $\sin \theta =\frac{y}{r}$ Substitute $-5$ for $y$ and $5\sqrt{2}$ for $r$. \begin{align} & \sin \theta =-\frac{5}{5\sqrt{2}} \\ & =-\frac{1}{\sqrt{2}} \\ & =-\frac{1}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}} \\ & =\frac{\sqrt{2}}{2} \end{align} Recall the trigonometric expression of $\cos \theta$. $\cos \theta =\frac{x}{r}$ Substitute $5$ for $x$ and $5\sqrt{2}$ for $r$. \begin{align} & \cos \theta =\frac{5}{5\sqrt{2}} \\ & =\frac{1}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}} \\ & =\frac{\sqrt{2}}{2} \end{align} Recall the trigonometric expression of $\tan \theta$. $\tan \theta =\frac{y}{x}$ Substitute $5$ for $x$ and $-5$ for $y$. \begin{align} & \tan \theta =\frac{-5}{5} \\ & =-1 \end{align} Recall the trigonometric expression of $\csc \theta$. $\csc \theta =\frac{r}{y}$ Substitute $-5$ for $y$ and $5\sqrt{2}$ for $r$. \begin{align} & \csc \theta =\frac{5\sqrt{2}}{-5} \\ & =-\sqrt{2} \end{align} Recall the trigonometric expression of $\sec \theta$. $\sec \theta =\frac{r}{x}$ Substitute $5$ for $x$ and $5\sqrt{2}$ for $r$. \begin{align} & \sec \theta =\frac{5\sqrt{2}}{5} \\ & =\sqrt{2} \end{align} Recall the trigonometric expression of $\cot \theta$. $\cot \theta =\frac{x}{y}$ Substitute $5$ for $x$ and $-5$ for $y$. \begin{align} & \cot \theta =\frac{5}{-5} \\ & =-1 \end{align} The six trigonometric functions of $\theta$ for point $\left( 5,-5 \right)$ are, $\sin \theta =-\frac{\sqrt{2}}{2},\cos \theta =\frac{\sqrt{2}}{2},\tan \theta =-1,\csc \theta =-\sqrt{2},\sec \theta =\sqrt{2}$ and $\cot \theta =-1$.