Problem 73 Find the signs of the six trigon... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 6: Problem 73

Find the signs of the six trigonometric function values for the given angles. $$319^{\circ}$$

Short Answer

Expert verified

In the fourth quadrant: $ \sin x $ is negative, $ \cos x $ is positive, $ \tan x $ is negative, $ \csc x $ is negative, $ \sec x $ is positive, $ \cot x $ is negative.

Step by step solution

- Determine the Quadrant

Find in which quadrant the angle lies. Since the angle is given as 319掳, subtract 360掳 to convert it within the standard range (0掳 to 360掳): 319掳 lies in the fourth quadrant (since 270掳 < 319掳 < 360掳).

- Recall Signs of Trigonometric Functions in Quadrant IV

In the fourth quadrant: - $ \sin x $ is negative- $ \cos x $ is positive- $ \tan x $ is negative- $ \csc x $ is negative (because it is the reciprocal of $ \sin x $)- $ \sec x $ is positive (because it is the reciprocal of $ \cos x $)- $ \cot x $ is negative (because it is the reciprocal of $ \tan x $)

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants. Each quadrant affects the signs of trigonometric functions differently. Quadrants are determined by the angle:

Quadrant I: 0掳 to 90掳
Quadrant II: 90掳 to 180掳
Quadrant III: 180掳 to 270掳
Quadrant IV: 270掳 to 360掳

For example, an angle of 319掳 lies in the fourth quadrant since it falls between 270掳 and 360掳. Understanding which quadrant an angle is in helps us determine the signs of the trigonometric functions associated with that angle.

Trigonometric Function Values

Trigonometric functions include sine ($ \sin $), cosine ($ \cos $), and tangent ($ \tan $), as well as their reciprocals. The values of these functions depend on the angle and the quadrant it lies in.
For example:

In Quadrant I, all trigonometric functions are positive.
In Quadrant II, $ \sin $ and $ \csc $ are positive, while $ \cos, \sec, \tan, $ and $ \cot $ are negative.
In Quadrant III, $ \tan $ and $ \cot $ are positive, while $ \sin, \csc, \cos, $ and $ \sec $ are negative.
In Quadrant IV, $ \cos $ and $ \sec $ are positive, while $ \sin, \csc, \tan, $ and $ \cot $ are negative.

In our case, with an angle of 319掳 in Quadrant IV, we will have:

$ \sin 319^\circ < 0 $
$ \cos 319^\circ > 0 $
$ \tan 319^\circ < 0 $

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are the inverses of sine, cosine, and tangent. They include cosecant ($ \csc $), secant ($ \sec $), and cotangent ($ \cot $).

$ \csc x = \frac{1}{\sin x} $
$ \sec x = \frac{1}{\cos x} $
$ \cot x = \frac{1}{\tan x} $

The signs of the reciprocals are directly related to the signs of their base trigonometric functions:

Since $ \sin 319^\circ < 0 \, \csc 319^\circ $ is also negative.
Since $ \cos 319^\circ > 0 \, \sec 319^\circ $ is also positive.
Since $ \tan 319^\circ < 0 \, \cot 319^\circ $ is also negative.

This makes it easier to determine the signs of these functions by first understanding the signs of $ \sin \, \cos \, $ and $ \tan $.

Negative and Positive Trigonometric Values

The signs of trigonometric functions are crucial in solving trigonometric problems. They indicate the direction of the function's value on the unit circle.
Here's a quick rundown of the signs in different quadrants:

Quadrant I: All positive
Quadrant II: $ \sin $ and $ \csc $ are positive
Quadrant III: $ \tan $ and $ \cot $ are positive
Quadrant IV: $ \cos $ and $ \sec $ are positive

By applying this to our original problem with an angle in Quadrant IV (319掳):

$ \sin 319^\circ < 0 $
$ \cos 319^\circ > 0 $
$ \tan 319^\circ < 0 $
$ \csc 319^\circ < 0 $
$ \sec 319^\circ > 0 $
$ \cot 319^\circ < 0 $

Remembering these patterns helps solve exercises about trigonometric function values efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the signs of the six trigonometric function values for the given angles. $$319^{\circ}$$

Short Answer

Step by step solution

- Determine the Quadrant

- Recall Signs of Trigonometric Functions in Quadrant IV

Key Concepts

Quadrants in Trigonometry

Trigonometric Function Values

Reciprocal Trigonometric Functions

Negative and Positive Trigonometric Values

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Geometry

Probability and Statistics

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.