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Found in: Page 522

### Essential Calculus: Early Transcendentals

Book edition 2nd
Author(s) James Stewart
Pages 830 pages
ISBN 9781133112280

# Identify the curve by finding a Cartesian equation for the curve.$${\rm{\theta = }}{{\rm{\pi }} \mathord{\left/ {\vphantom {{\rm{\pi }} 3}} \right. \kern-\nulldelimiterspace} 3}$$

The curve is$${\rm{y = }}\sqrt {\rm{3}} {\rm{x}}$$.

See the step by step solution

## Step 1: Find a Cartesian equation for the curve and use it to identify it.

Given: $${\rm{\theta = }}\frac{{\rm{\pi }}}{{\rm{3}}}$$

Take the tan from both sides and combine them.

$${\rm{tan\theta = tan}}\frac{{\rm{\pi }}}{{\rm{3}}}$$

Known value, $${\rm{tan\theta = }}\frac{{{\rm{sin\theta }}}}{{{\rm{cos\theta }}}}$$

$$\frac{{{\rm{sin\theta }}}}{{{\rm{cos\theta }}}}{\rm{ = }}\sqrt {\rm{3}}$$

Both the numerator and denominator should be multiplied by$${\rm{r}}$$.

$$\frac{{{\rm{rsin\theta }}}}{{{\rm{rcos\theta }}}}{\rm{ = }}\sqrt {\rm{3}}$$

Keep in mind that: $${\rm{rcos\theta = x\& rsin\theta = y}}$$

\begin{aligned}{l}\frac{{\rm{y}}}{{\rm{x}}}{\rm{ = }}\sqrt {\rm{3}} \\{\rm{y = }}\sqrt {\rm{3}} {\rm{x}}\end{aligned}

## Step 2: Result.

Therefore, the curve is$${\rm{y = }}\sqrt {\rm{3}} {\rm{x}}$$.