Q. 13 In Problems 13鈥�20, P=x,y聽is t... [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. 13 (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 $t$ .
$(\frac{\sqrt{3}}{2}, \frac{1}{2})$ .

Short Answer

Expert verified

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

$\begin{array}{rcl} \sin t & = & \frac{1}{2} \\ \cos t & = & \frac{\sqrt{3}}{2} \\ \tan t & = & \frac{1}{3} \sqrt{3} \\ \csc t & = & 2 \\ \sec t & = & \frac{2}{3} \sqrt{3} \\ \cot t & = & \sqrt{3} \end{array}$

Step by step solution

Given information.

The point on the unit circle that corresponds to $t$ is $P (x, y) = (\frac{\sqrt{3}}{2}, \frac{1}{2})$ .

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:

Substitute $x = \frac{\sqrt{3}}{2}, y = \frac{1}{2}$ in the six trigonometric functions and simplify.

Find $\sin t, \cos t, \tan t$ .

$\begin{array}{rcl} \sin t & = & y \\ \sin t & = & \frac{1}{2} \end{array}$

$\begin{array}{rcl} \cos t & = & x \\ \cos t & = & \frac{\sqrt{3}}{2} \end{array}$

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

Evaluate

Find $\csc t, \sec t, \cot t$ .

role="math" localid="1646310370849" $c s c t = \frac{1}{y} c s c t = \frac{1}{\frac{1}{2}} c s c t = 2$

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

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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 $t$ .
$(\frac{\sqrt{3}}{2}, \frac{1}{2})$ .

Short Answer

Step by step solution

Given information.

Definitions.

Evaluate:

Evaluate

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91影视

In Problems 13鈥�20, P=x,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.32,12.

Short Answer

Step by step solution

Given information.

Definitions.

Evaluate:

Evaluate

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Discrete Mathematics

Logic and Functions

Mechanics Maths

Pure Maths

Theoretical and Mathematical Physics

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 $t$ .
$(\frac{\sqrt{3}}{2}, \frac{1}{2})$ .