/*! 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 39 Solve the following equations. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 39

Solve the following equations. $$\sin ^{2} \theta=\frac{1}{4}, 0 \leq \theta<2 \pi$$

Short Answer

Expert verified
Question: Solve the trigonometric equation \(\sin^2\theta = \frac{1}{4}\) for all angles \(\theta\) in the interval \(0 \leq \theta < 2\pi\). Answer: The complete solution for \(\theta\) in the given interval is \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}\).

Step by step solution

01

Take the square root of both sides

In order to solve for \(\theta\), we need to get rid of the exponent 2 on the sine function. We can do this by taking the square root of both sides of the equation: $$\sqrt{\sin^2\theta} = \sqrt{\frac{1}{4}}$$ This gives us: $$\sin\theta = \pm\frac{1}{2}$$
02

Determine the angles for the positive sine value

We now need to find the angles in the interval \(0 \leq \theta < 2\pi\) where the sine function equals \(\pm\frac{1}{2}\). Let's start with the positive sine value: $$\sin\theta = \frac{1}{2}$$ Recall that the sine function is positive in the first and second quadrants. The reference angle where the sine has a value of \(\frac{1}{2}\) is the angle \(\frac{\pi}{6}\), so our solutions in the first and second quadrants are: $$\theta_1 = \frac{\pi}{6}\ \ , \ \theta_2 = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$$
03

Determine the angles for the negative sine value

Now, let's find the angles where the sine function has a value of \(-\frac{1}{2}\) in our interval: $$\sin\theta = -\frac{1}{2}$$ The sine function is negative in the third and fourth quadrants. The reference angle where the sine has a value of \(\frac{1}{2}\) is the angle \(\frac{\pi}{6}\), so our solutions in the third and fourth quadrants are: $$\theta_3 = \pi + \frac{\pi}{6} = \frac{7\pi}{6}\ \ , \ \theta_4 = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}$$
04

State the complete solution

Now that we have found all the angles for both the positive and negative sine values in the given interval, we can state the complete solution for \(\theta\): $$\theta = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$$

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

One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. $$\sin \theta=-\frac{4}{5} \text { and } \pi<\theta<3 \pi / 2$$

Graphing sine and cosine functions Beginning with the graphs of \(y=\sin x\) or \(y=\cos x,\) use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. $$g(x)=-2 \cos (x / 3)$$

Optimal boxes Imagine a lidless box with height \(h\) and a square base whose sides have length \(x\). The box must have a volume of \(125 \mathrm{ft}^{3}\). a. Find and graph the function \(S(x)\) that gives the surface area of the box, for all values of \(x>0\) b. Based on your graph in part (a), estimate the value of \(x\) that produces the box with a minimum surface area.

Parabola properties Consider the general quadratic function \(f(x)=a x^{2}+b x+c,\) with \(a \neq 0\). a. Find the coordinates of the vertex in terms of \(a\). \(b\), and \(c\). b. Find the conditions on \(a, b,\) and \(c\) that guarantee that the graph of \(f\) crosses the \(x\) -axis twice.

Suppose the probability of a server winning any given point in a tennis match is a constant \(p,\) with \(0 \leq p \leq 1\).Then the probability of the server winning a game when serving from deuce is $$f(p)=\frac{p^{2}}{1-2 p(1-p)}$$,a. Evaluate \(f(0.75)\) and interpret the result. b. Evaluate \(f(0.25)\) and interpret the result. (Source: The College Mathematics Journal 38, 1, Jan 2007).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Geometry

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.