/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 Find the exact value of the trig... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 20

Find the exact value of the trigonometric functions at the indicated angle. \(\sin x, \cot x\), and \(\csc x\) for \(x=5 \pi / 6\)

Short Answer

Expert verified
For \(x = \frac{5\pi}{6}\), the exact values of the trigonometric functions are: \[\sin(x) = \frac{1}{2}, \quad \cot(x) = -\sqrt{3}, \quad \text{and} \quad \csc(x) = 2.\]

Step by step solution

01

Identify the Position of the Angle on the Unit Circle

Let's first place the angle \(x = \frac{5\pi}{6}\) on the unit circle, so we can identify its reference angle and quadrant. The angle \(\frac{5\pi}{6}\) is equal to \(\frac{5(180^\circ)}{6} = 150^\circ\). Since the angle is between \(90^\circ\) and \(180^\circ\), it lies in the second quadrant.
02

Get the Reference Angle

Now let's find the reference angle by subtracting the angle between the second quadrant and the x-axis from the given angle: \( \alpha = 180^\circ - 150^\circ = 30^\circ \). The reference angle between the terminal side and the x-axis is \(\alpha = 30^\circ\). This reference angle has special properties, and we can use it to find the unknown trigonometric functions.
03

Find \(\sin(x)\) using the Reference Angle

In a 30-60-90 reference triangle, the sides have a ratio of 1 : 2 : \(\sqrt{3}\), where 1 is the length of the side opposite the 30-degree angle, 2 is the length of the hypotenuse, and \(\sqrt{3}\) is the length of the side opposite the 60-degree angle. From the definition of sine, we have: \( \sin(\alpha) = \frac{\text{opposite side}}{\text{hypotenuse}} \). So, for a 30-degree angle, \( \sin(30^\circ) = \frac{1}{2} \). Since the angle \(x\) is in the second quadrant, the \(x\)-coordinate is negative while the \(y\)-coordinate is positive. Therefore, \(\sin(x) = \sin(\frac{5\pi}{6}) = \frac{1}{2}\).
04

Find \(\cot(x)\) using the Reference Angle

To find the \(\cot(x)\), let's first find \(\tan(x)\), since \(\cot(x) = \frac{1}{\tan(x)}\). In the 30-60-90 reference triangle, using the definition of tangent, we have: \( \tan(\alpha) = \frac{\text{opposite side}}{\text{adjacent side}} \). So, for a 30-degree angle, \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \). Since the angle \(x\) is in the second quadrant, the tangent function is negative. Therefore, \(\tan(x) = -\tan(\frac{5\pi}{6}) = -\frac{1}{\sqrt{3}}\). Now, since \(\cot(x) = \frac{1}{\tan(x)}\), we can divide 1 by the tangent value: \( \cot(x) = -\sqrt{3} \).
05

Find \(\csc(x)\) using the Value of \(\sin(x)\)

Since the cosecant function is the reciprocal of the sine function, we can easily find it as: \( \csc(x) = \frac{1}{\sin(x)} \). Using the sine value we calculated in Step 3: \( \csc(x) = \csc(\frac{5\pi}{6}) = \frac{1}{\frac{1}{2}} = 2 \). Finally, we have found the exact trigonometric values: \(\sin(x) = \frac{1}{2}\), \(\cot(x) = -\sqrt{3}\), and \(\csc(x) = 2\).

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!

One App. One Place for Learning.

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

Get started for free

Most popular questions from this chapter

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=x^{2} \sin \frac{1}{x} $$

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ f(x)=\sqrt[3]{x}-\sqrt[3]{x+1} $$

Let \(f(x)=x+\frac{1}{100} \sin 100 x\) a. Plot the graph of \(f\) using the viewing window \([-10,10] \times[-10,10]\) b. Plot the graph of \(f\) using the viewing window \([-0.1,0.1] \times[-0.1,0.1]\) c. Explain why the two displays obtained in parts (a) and (b) taken together give a complete description of the graph of \(f\).

Plot the graph of the function \(f\) in an appropriate viewing window. (Note: The answer is not unique.) $$ \begin{array}{l} f(x)=-2 x^{4}+5 x^{2}-4\\\ \text { 7. } f(x)=\frac{x^{3}}{x^{3}+1} \end{array} $$

Let \(f\) be a function defined by \(f(x)=\sqrt{x}+\sin x\) on the interval \([0,2 \pi]\). a. Find an even function \(g\) defined on the interval \([-2 \pi, 2 \pi]\) such that \(g(x)=f(x)\) for all \(x\) in \([0,2 \pi]\). b. Find an odd function \(h\) defined on the interval \([-2 \pi, 2 \pi]\) such that \(h(x)=f(x)\) for all \(x\) in \([0,2 \pi]\).
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.