Q. 14 In Problems 13鈥�20, Px,yis the ... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 6: Q. 14 (page 380)

In Problems 13鈥�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 $t$ .
$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ .

Short Answer

Expert verified

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

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

Step by step solution

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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91影视

In Problems 13鈥�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 $t$ .
$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ .

Short Answer

Step by step solution

Definitions.

Evaluate

Evaluate.

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91影视

In Problems 13鈥�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.

Short Answer

Step by step solution

Definitions.

Evaluate

Evaluate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Theoretical and Mathematical Physics

Decision Maths

Pure Maths

Geometry

Study anywhere. Anytime. Across all devices.

In Problems 13鈥�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 $t$ .
$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ .