Book edition 6th

Author(s) Sullivan

Pages 1200 pages

ISBN 9780321795465

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Short Answer

In Problems 13–20, $P (x, y)$ is the point on the unit circle that corresponds to a real number $t$ . Find the exact values of the six trigonometric functions of $t$ .
$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ .

The exact values of the six trigonometric functions of $t$ for $P = (\frac{1}{2}, - \frac{\sqrt{3}}{2})$ are:

$\begin{matrix} \sin t = - \frac{\sqrt{3}}{2} \\ \cos t = \frac{1}{2} \\ \tan t = - \sqrt{3} \\ \csc t = - \frac{2}{3} \sqrt{3} \\ \sec t = 2 \\ \cot t = - \frac{1}{3} \sqrt{3} \end{matrix}$

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Definitions.

For a point $P (x, y)$ on a unit circle that corresponds to $t$ , the six trigonometric functions are defined as follows:

$\begin{matrix} \sin t = y \\ \cos t = x \\ \tan t = \frac{y}{x} \\ \csc t = \frac{1}{y} \\ \sec t = \frac{1}{x} \\ \cot t = \frac{x}{y} \end{matrix}$

Evaluate

Here, $P (x, y) = (\frac{1}{2}, - \frac{\sqrt{3}}{2})$ .

Substitute these values in the six trigonometric functions and simplify.

Find $\sin t$ .

$\begin{array}{rcl} \sin t & = & y \\ \sin t & = & - \frac{\sqrt{3}}{2} \end{array}$

Find role="math" localid="1646311483257" $\cos t$ .

$\begin{array}{rcl} \cos t & = & x \\ \cos t & = & \frac{1}{2} \end{array}$

Find $\tan t$ .

$\begin{array}{rcl} \tan t & = & \frac{y}{x} \\ \tan t & = & \frac{- \frac{\sqrt{3}}{2}}{\frac{1}{2}} \\ \tan t & = & - \sqrt{3} \end{array}$

Evaluate.

Find $c s c t$ .

$\begin{array}{rcl} c s c t & = & \frac{1}{y} \\ c s c t & = & \frac{1}{\frac{- \sqrt{3}}{2}} \\ c s c t & = & - \frac{2}{\sqrt{3}} \\ c s c t & = & - \frac{2}{3} \sqrt{3} \end{array}$

Find $s e c t$ .

$s e c t = \frac{1}{x} s e c t = \frac{1}{\frac{1}{2}} s e c t = 2$

Find $c o t t$ .

$\begin{array}{rcl} c o t t & = & \frac{x}{y} \\ c o t t & = & \frac{\frac{1}{2}}{- \frac{\sqrt{3}}{2}} \\ c o t t & = & - \frac{1}{\sqrt{3}} \\ c o t t & = & - \frac{1}{3} \sqrt{3} \end{array}$

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Precalculus Enhanced with Graphing Utilities

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Short Answer

In Problems 13–20, $P (x, y)$ is the point on the unit circle that corresponds to a real number $t$ . Find the exact values of the six trigonometric functions of $t$ .
$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ .

Step by Step Solution

Definitions.

Evaluate

Evaluate.

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Precalculus Enhanced with Graphing Utilities

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Short Answer

In Problems 13–20, Px,yis the point on the unit circle that corresponds to a real number t. Find the exact values of the six trigonometric functions of t.12,-32.

Step by Step Solution

Definitions.

Evaluate

Evaluate.

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In Problems 13–20, $P (x, y)$ is the point on the unit circle that corresponds to a real number $t$ . Find the exact values of the six trigonometric functions of $t$ .
$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ .