/*! 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 66 If \(270^{\circ}<\theta<36... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 66

If \(270^{\circ}<\theta<360^{\circ}\), then which quadrant does \(\frac{\theta}{2}\) terminate in? a. \(\mathrm{QI}\) b. QII c. QIII d. QIV

Short Answer

Expert verified
The answer is b. QII.

Step by step solution

01

Identify the range of \( \theta \)

Given that \( 270^{\circ} < \theta < 360^{\circ} \). This means \( \theta \) is in the fourth quadrant. We need to determine the effects on \( \theta \) when divided by two.
02

Divide the inequality by 2

To find \( \frac{\theta}{2} \), divide the entire inequality by 2: \( \frac{270^{\circ}}{2} < \frac{\theta}{2} < \frac{360^{\circ}}{2} \). This simplifies to \( 135^{\circ} < \frac{\theta}{2} < 180^{\circ} \).
03

Determine quadrant based on \( \frac{\theta}{2} \) range

The range \( 135^{\circ} < \frac{\theta}{2} < 180^{\circ} \) falls within the second quadrant, which spans from \( 90^{\circ} \) to \( 180^{\circ} \).
04

Confirm which option matches the quadrant

The second quadrant is represented by option b, QII.

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.

Angle Measurement
In trigonometry, understanding angle measurement is a fundamental aspect. Angles can be measured in various ways, including degrees and radians. In this context, we focus on degrees.
  • A full circle is composed of 360 degrees.
  • Each quadrant of the circle is equal to 90 degrees.
  • Angles are often defined in a range, such as \(0^{\circ} < \theta < 360^{\circ}\), to represent their possible locations on a coordinate plane.
To understand angle measurement more clearly, consider that if \(\theta\) represents an angle, then understanding how dividing this angle by a certain number affects its position is crucial, like in our exercise. The main idea is to comprehend how transformations, such as dividing by two, can potentially change the quadrant in which the angle terminates.
Quadrants in Trigonometry
The coordinate plane is divided into four quadrants, each corresponding to a specific range of angles.
  • **Quadrant I (QI):** This quadrant contains angles from \(0^{\circ}\) up to \(90^{\circ}\). Positive values for both sine and cosine functions are typical here.
  • **Quadrant II (QII):** Angles from \(90^{\circ}\) to \(180^{\circ}\) are in the second quadrant. The sine function is positive, while the cosine function is negative.
  • **Quadrant III (QIII):** This quadrant includes angles from \(180^{\circ}\) to \(270^{\circ}\). Here, both sine and cosine functions are negative.
  • **Quadrant IV (QIV):** Finally, angles from \(270^{\circ}\) to \(360^{\circ}\) belong here. In this quadrant, cosine is positive, while sine is negative.
Recognizing which quadrant an angle falls into is necessary to solve trigonometric problems, as it provides insights into the signs and values of trigonometric functions. Particularly, if we take an angle from one quadrant and transform it (such as halving, like in our exercise), it may end up in another quadrant.
Inequality Solving
Solving inequalities is a foundational skill in mathematics, including trigonometry. Inequalities demonstrate a range of possible values that satisfy a particular condition. To illustrate:
  • If you have an inequality, such as \(270^{\circ} < \theta < 360^{\circ}\), it implies that \(\theta\) lies within a specific segment of a circle.
  • Solving inequalities can involve transforming or simplifying expressions while maintaining the inequality's truth. For example, in this exercise, dividing the entire inequality by 2 helps us understand where \(\frac{\theta}{2}\) terminates.
The result is another inequality that removes any ambiguity about the placement of the angle's transformation. As a result, by identifying the new range, we are able to conclude that \(\frac{\theta}{2}\) fits within the second quadrant.

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

Use exact values to show that each of the following is true. $$ \text { If } \tan A=\frac{5}{12}, \text { find } \tan 2 A $$

For Questions 1 through 3, complete each double-angle identity. \(\sin 2 x=\)

Let \(\tan \theta=3\) with \(\theta\) in QI and find the following. $$ \cos 2 \theta $$

Simplify each of the following. $$ \cos ^{2} 165^{\circ}-\sin ^{2} 165^{\circ} $$

Prove each of the following identities. $$ \cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.