/*! 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 79 Find all the values of \(\theta(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 79

Find all the values of \(\theta(\theta\) is any real number) for which the equation \(3 \csc \theta=4 \sin \theta\) is true.

Short Answer

Expert verified
\(\theta = \frac{\pi}{3} + 2k\pi, \frac{2\pi}{3} + 2k\pi, \frac{4\pi}{3} + 2k\pi, \frac{5\pi}{3} + 2k\pi\).

Step by step solution

01

Recall Definitions and Pythagorean Identity

The equation given is \(3 \csc \theta = 4 \sin \theta\). Start by recalling the definition of cosecant: \( \csc \theta = \frac{1}{\sin \theta} \). Thus, the equation becomes \(3 \cdot \frac{1}{\sin \theta} = 4 \sin \theta\). Rewriting gives us \(3 = 4 \sin^2 \theta\).
02

Rearrange and Solve the Equation

Rearrange the equation: \(4 \sin^2 \theta = 3\). Then, divide both sides by 4 to isolate \(\sin^2 \theta\): \(\sin^2 \theta = \frac{3}{4}\).
03

Take the Square Root

Take the square root of both sides to solve for \(\sin \theta\): \(\sin \theta = \pm \sqrt{\frac{3}{4}}\). This simplifies to \(\sin \theta = \pm \frac{\sqrt{3}}{2}\).
04

Find General Solutions

Recall the angles where \(\sin \theta = \pm \frac{\sqrt{3}}{2}\). From the unit circle, \(\sin \theta = \frac{\sqrt{3}}{2}\) at \(\theta = \frac{\pi}{3} + 2k\pi\) and \(\theta = \frac{2\pi}{3} + 2k\pi\), and \(\sin \theta = -\frac{\sqrt{3}}{2}\) at \(\theta = \frac{4\pi}{3} + 2k\pi\) and \(\theta = \frac{5\pi}{3} + 2k\pi\) for any integer \(k\).

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.

Cosecant
When working with trigonometric equations, understanding the cosecant function is important. Cosecant, denoted as \( \csc \theta \), is a trigonometric function equivalent to the reciprocal of the sine function.
This means that \( \csc \theta = \frac{1}{\sin \theta} \).
The cosecant function is undefined wherever the sine function is zero, which happens at angles like \( 0, \pi, 2\pi, \ldots \).
  • Cosecant is useful for solving equations where a reciprocal is needed.
  • It's important to remember that \( \csc \theta \) is undefined at points where \( \sin \theta = 0 \).
Understanding \( \csc \theta \) helps in tackling equations like \( 3 \csc \theta = 4 \sin \theta \), since we can convert it to a form involving only the sine function, making the equation easier to solve.
Unit Circle
The unit circle is a fundamental tool in trigonometry, especially when dealing with sine and its reciprocal, cosecant. It is a circle with a radius of 1, centered at the origin of a coordinate plane.
Each angle on the unit circle can provide valuable information about the sine, cosine, and other trigonometric functions.
  • For any angle \( \theta \), \( \sin \theta \) is represented by the y-coordinate of the point where the terminal side of the angle intersects the unit circle.
  • The x-coordinate at this intersection gives \( \cos \theta \).
For example:
  • \( \sin \theta = \frac{\sqrt{3}}{2} \) happens at \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \).
  • \( \sin \theta = -\frac{\sqrt{3}}{2} \) is found at \( \theta = \frac{4\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).
Knowing these values allows us to determine solutions for trigonometric equations using symmetry and periodicity of the circle.
Pythagorean Identity
The Pythagorean identity is a critical concept used in solving trigonometric equations. This identity is: \[ \sin^2 \theta + \cos^2 \theta = 1 \].
This fundamental identity stems from the Pythagorean theorem in a unit circle.
  • It helps in expressing one trigonometric function in terms of another.
  • This identity can simplify expressions and solve equations involving \( \sin \) or \( \cos \).
In the exercise \( 3 \csc \theta = 4 \sin \theta \):
  • We use the identity indirectly by focusing on \( \sin \theta \) since \( \csc \theta \) is \( \frac{1}{\sin \theta} \).
  • The rearranging step leads us to \( 4 \sin^2 \theta = 3 \), helping us isolate and solve for \( \sin \theta \).
It is essential to recognize how the Pythagorean identity underpins many trigonometric principles, including the relation between \( \sin \) and \( \cos \), and it's pivotal in breaking down more complex facets of trigonometry.

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

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=-\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)-2 \cos \left(x-\frac{\pi}{6}\right),-\pi \leq x \leq \pi$$

Determine whether each statement is true or false. Is cosecant an even or an odd function? Justify your answer.

Solve the equation \(\tan (2 x-\pi)=0\) for \(x\) in the interval \([-\pi, \pi]\) by graphing.

In Exercises \(67-94,\) add the ordinates of the individual functions to graph each summed function on the indicated interval. $$y=\frac{1}{2} \sin x+2 \cos (4 x),-\pi \leq x \leq \pi$$

Determine whether each statement is true or false. \(\cos (2 n \pi+\theta)=\cos \theta, n\) an integer.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

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.