/*! 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 3 Sketch the following angles. \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 3

Sketch the following angles. \(-\boldsymbol{\pi}\)

Short Answer

Expert verified
Starting from the positive x-axis, a clockwise half circle rotation lands at the negative x-axis.

Step by step solution

01

Understand the Angle

The given angle is -蟺. In radians, 蟺 is the angle for half a circle, or 180 degrees. -蟺 means we need to rotate in the negative direction which is clockwise.
02

Draw the Coordinate System

Draw a coordinate system with the x-axis (horizontal) and y-axis (vertical). This will help visualize the angle correctly.
03

Locate the Angle

Start from the positive x-axis (0 radians) and rotate clockwise. Since it is -蟺, we need to rotate half a circle. This means we will end up on the negative x-axis.
04

Mark the Angle

Draw an angle starting from the positive x-axis and ending at the negative x-axis through a clockwise rotation. Label this angle as -蟺.

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.

coordinate system
To properly sketch angles, it's essential to first understand the coordinate system. The Cartesian coordinate system consists of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin, \((0, 0)\).

When sketching angles, you start from the positive x-axis, which points to the right. Movements counterclockwise from the positive x-axis indicate positive angles, while movements clockwise indicate negative angles. This framework is crucial for visualizing angles correctly, including those measured in radians.
negative rotation
In trigonometry, angles can have positive or negative values. Negative rotation means moving in a clockwise direction from the starting point on the coordinate system.

For instance, to sketch \(-\pi\) radians, you start at the positive x-axis (0 radians) and move clockwise, covering half of the circle. This would place the terminal side of the angle along the negative x-axis. Understanding this concept is key to accurately drawing angles represented by negative radians.
angle sketching
Sketching angles involves a few methodical steps to ensure precision.
  • Step 1: Draw the coordinate system with both x and y axes.
  • Step 2: Identify the starting point which is usually the positive x-axis (0 radians).
  • Step 3: Determine the direction of rotation needed (clockwise for negative angles, counterclockwise for positive).
  • Step 4: Rotate by the given angle measurement and mark the terminal side of the angle.

For example, sketching \(-\pi\) would involve rotating clockwise from the positive x-axis to the negative x-axis, effectively marking a half-circle turn.
蟺 radians
\pi\ radians is a fundamental unit in measuring angles, equivalent to 180 degrees. Radians and degrees are both units for measuring angles, but radians offer a more natural fit for many mathematical calculations.

A full circle measures \(2\pi\) radians, and a half-circle is \pi\ radians.
Thus, when we talk about \(-\pi\) radians, we mean a half-circle rotation in the clockwise direction.

It's crucial to recognize that radians provide a straightforward way to express various circular and trigonometric concepts, and understanding \pi\ is vital for working with angles effectively.

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

Amplitude and Mid-Line. Consider the function \(g(t)=a \sin b t+k\), where \(a\) and \(b\) are positive. a) Show that the maximum and minimum of \(g(t)\) are \(k+a\) and \(k-a\), respectively. b) Show that the mid-line is the line \(y=k\). c) Show that the amplitude of \(g(t)\) is \(a\).

Use a calculator to find the values of the following trigonometric functions. \(\tan (\pi / 7)\)

On a 5 -mi stretch of highway, the road decreases in elevation at an angle of \(4^{\circ} .\) How much lower is a car after traveling on this part of the highway? (Remember that there are 5280 feet in a mile.)

Graph the following functions on \([-2 \pi, 2 \pi]\). a) \(y=3 \sin t+4\) b) \(y=3 \sin (t-\pi / 4)+4\) c) How does the horizontal shift change the graph?

Use a calculator to evaluate the following trigonometric functions. $$ \sec 286^{\circ} $$
See all solutions

Recommended explanations on Biology Textbooks

Cell Cycle

Read Explanation

Cells

Read Explanation

Organ Systems

Read Explanation

Biological Organisms

Read Explanation

Plant Biology

Read Explanation

Cellular Energetics

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.