## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 4 - Section 4.4 - Trigonometric Functions of Any Angle - Exercise Set - Page 575: 4

#### Answer

The six trigonometric functions of $\theta$ for point $\left( 3,7 \right)$ are, $\sin \theta =\frac{7\sqrt{58}}{58},\cos \theta =\frac{3\sqrt{58}}{58},\tan \theta =\frac{7}{3},\csc \theta =\frac{\sqrt{58}}{7},\sec \theta =\frac{\sqrt{58}}{3}$ and $\cot \theta =\frac{3}{7}$.

#### Work Step by Step

Consider the point $\left( 3,7 \right)$. Here, $x=3$ and $y=7$ The six trigonometric functions of $\theta$ are defined in the term of ratios. According to the Pythagoras theorem, the hypotenuse is, $r=\sqrt{{{x}^{2}}+{{y}^{2}}}$ Substitute $3$ for $x$ and $7$ for $y$. \begin{align} & r=\sqrt{{{\left( 3 \right)}^{2}}+{{\left( 7 \right)}^{2}}} \\ & =\sqrt{9+49} \\ & =\sqrt{58} \end{align} Write the trigonometric expression of $\sin \theta$. $\sin \theta =\frac{y}{r}$ Substitute $7$ for $y$ and $\sqrt{58}$ for $r$. \begin{align} & \sin \theta =\frac{7}{\sqrt{58}} \\ & =\frac{7}{\sqrt{58}}.\frac{\sqrt{58}}{\sqrt{58}} \\ & =\frac{7\sqrt{58}}{58} \end{align} Recall the trigonometric expression of $\cos \theta$. $\cos \theta =\frac{x}{r}$ Substitute $3$ for $x$ and $\sqrt{58}$ for $r$. \begin{align} & \cos \theta =\frac{3}{\sqrt{58}} \\ & =\frac{3}{\sqrt{58}}.\frac{\sqrt{58}}{\sqrt{58}} \\ & =\frac{3\sqrt{58}}{58} \end{align} Recall the trigonometric expression of $\tan \theta$. $\tan \theta =\frac{y}{x}$ Substitute $3$ for $x$ and $7$ for $y$. $\tan \theta =\frac{7}{3}$ Recall the trigonometric expression of $\csc \theta$. $\csc \theta =\frac{r}{y}$ Substitute $7$ for $y$ and $\sqrt{58}$ for $r$. $\csc \theta =\frac{\sqrt{58}}{7}$ Recall the trigonometric expression of $\sec \theta$. $\sec \theta =\frac{r}{x}$ Substitute $3$ for $x$ and $\sqrt{58}$ for $r$. $\sec \theta =\frac{\sqrt{58}}{3}$ Recall the trigonometric expression of $\cot \theta$. $\cot \theta =\frac{x}{y}$ Substitute $3$ for $x$ and $7$ for $y$. $\cot \theta =\frac{3}{7}$ Thus, the six trigonometric functions of $\theta$ for point $\left( 3,7 \right)$ are, $\sin \theta =\frac{7\sqrt{58}}{58},\cos \theta =\frac{3\sqrt{58}}{58},\tan \theta =\frac{7}{3},\csc \theta =\frac{\sqrt{58}}{7},\sec \theta =\frac{\sqrt{58}}{3}$ and $\cot \theta =\frac{3}{7}$.

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