Q. 83 A point on the terminal side of ... [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. 83 (page 381)

A point on the terminal side of an angle $胃$ in standard position is given. Find the exact value of each of the six trigonometric functions of $胃$ : $(\frac{1}{3}, \frac{1}{4})$

Short Answer

Expert verified

The exact value of the six trigonometric functions are,

$\sin 胃 = \frac{3}{5} \cos 胃 = \frac{4}{5} \tan 胃 = \frac{3}{4} \csc 胃 = \frac{5}{3} \sec 胃 = \frac{5}{4} \cot 胃 = \frac{4}{3}$

Step by step solution

Step 1. Given information

We are given a point $(\frac{1}{3}, \frac{1}{4})$

We need to find the exact value of each of the six trigonometric functions.

Step 2. Concept

Let $(x, y)$ be a point on the terminal side of the angle $胃$ in the standard position, which is also on the circle $x^{2} + y^{2} = r^{2}$ . Then,

localid="1646759908150" $\sin 胃 = \frac{y}{r} \cos 胃 = \frac{x}{r} \tan 胃 = \frac{y}{x} \csc 胃 = \frac{r}{y} \sec 胃 = \frac{r}{x} \cot 胃 = \frac{x}{y}$

Step 3. Finding the value of r

The given point is $(\frac{1}{3}, \frac{1}{4})$ . Here, $x = \frac{1}{3}, y = \frac{1}{4}$

To find $r$ , we have

$r = \sqrt{x^{2} + y^{2}} r = \sqrt{{(\frac{1}{3})}^{2} + {(\frac{1}{4})}^{2}} r = \sqrt{\frac{1}{9} + \frac{1}{16}} r = \sqrt{\frac{25}{144}} r = \frac{5}{12}$

Step 4. Finding the values of the six trigonometric functions

We are given a point $(\frac{1}{3}, \frac{1}{4})$ . Here, $x = \frac{1}{3}, y = \frac{1}{4}$ . Aslo, we have that $r = \frac{5}{12}$

$\sin 胃 = \frac{y}{r} = \frac{\frac{1}{4}}{\frac{5}{12}} = \frac{12}{20} = \frac{3}{5} \cos 胃 = \frac{x}{r} = \frac{\frac{1}{3}}{\frac{5}{12}} = \frac{12}{15} = \frac{4}{5} \tan 胃 = \frac{y}{x} = \frac{\frac{1}{4}}{\frac{1}{3}} = \frac{3}{4} \csc 胃 = \frac{r}{y} = \frac{\frac{5}{12}}{\frac{1}{4}} = \frac{20}{12} = \frac{5}{3} \sec 胃 = \frac{r}{x} = \frac{\frac{5}{12}}{\frac{1}{3}} = \frac{15}{12} = \frac{5}{4} \cot 胃 = \frac{x}{y} = \frac{\frac{1}{3}}{\frac{1}{4}} = \frac{4}{3}$

Step 5. Final Answer

The values of the six trigonometric functions of $胃$ ,

$\sin 胃 = \frac{3}{5} \cos 胃 = \frac{4}{5} \tan 胃 = \frac{3}{4} \csc 胃 = \frac{5}{3} \sec 胃 = \frac{5}{4} \cot 胃 = \frac{4}{3}$

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91影视

A point on the terminal side of an angle $胃$ in standard position is given. Find the exact value of each of the six trigonometric functions of $胃$ : $(\frac{1}{3}, \frac{1}{4})$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Concept

Step 3. Finding the value of r

Step 4. Finding the values of the six trigonometric functions

Step 5. Final Answer

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91影视

A point on the terminal side of an angle 胃in standard position is given. Find the exact value of each of the six trigonometric functions of胃:13,14

Short Answer

Step by step solution

Step 1. Given information

Step 2. Concept

Step 3. Finding the value of r

Step 4. Finding the values of the six trigonometric functions

Step 5. Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Applied Mathematics

Calculus

Logic and Functions

Probability and Statistics

Study anywhere. Anytime. Across all devices.

A point on the terminal side of an angle $胃$ in standard position is given. Find the exact value of each of the six trigonometric functions of $胃$ : $(\frac{1}{3}, \frac{1}{4})$