Problem 17 Construct angles with the follow... [FREE SOLUTION]

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Calculus and Its Applications

Larry J. Goldstein, David I. Schneider, Nakhle H. Asmar

$Math Studyset 91影视 Explanations$ Math

13 Edition

Chapter 8: Problem 17

Construct angles with the following radian measure. $\pi / 6,-2 \pi / 3,-\pi$

Short Answer

Expert verified

30 degrees counter-clockwise from x-axis, 120 degrees clockwise from x-axis, and 180 degrees clockwise from x-axis.

Step by step solution

Construct $\frac{\pi}{6}$

To construct an angle of $\pi/6$ radians, start by remembering that $\pi$ radians is equivalent to 180 degrees. Therefore, $\pi/6$ is $180/6 = 30$ degrees. Draw a 30-degree angle starting from the positive x-axis.

Construct $-\frac{2\pi}{3}$

For $-2\pi/3$ radians, first find the positive equivalent by considering $2\pi/3$, which converts to $2 \times 180/3 = 120$ degrees. Since the angle is negative, measure 120 degrees clockwise from the positive x-axis.

Construct $-\pi$

An angle of $-\pi$ radians is just $-180$ degrees. Measure 180 degrees clockwise from the positive x-axis. This will land on the negative x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

headline of the respective core concept

Understanding how to construct angles in radians is essential for mastering trigonometry and other math-related fields. This involves using simple tools like a compass and a protractor, and understanding the conversion between degrees and radians. Let's explore the main concepts: angle construction, radian measure, degree conversion, and negative angles.

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Angle construction is the method used to create angles of a specific measure. To begin constructing an angle, you usually start from the positive x-axis. Using a protractor, you mark off the desired angle in degrees. For angles given in radians, convert them to degrees first. Draw a ray from the origin of the x-axis that matches the protractor's angle measurement.

For example, to construct an angle of $\pi/6$ (which converts to 30 degrees), mark a 30-degree angle and draw your ray, forming the required angle.

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Radian measure is a way to express angles using the radius of a circle. One radian is the angle created when the arc length is equal to the radius of the circle. Unlike degrees, radians provide a more natural way to describe angles in terms of $\pi$.

Think about the circle's circumference, which is $2\pi r$. Splitting this into equal parts around the circle helps understand radian measures. For instance, a circle divided into six equal parts has angles of $\pi/6$, equivalent to 30 degrees. Using radians can simplify many trigonometric equations and calculations.

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Degree conversion is necessary to switch between degrees and radians, facilitating easier angle construction. To convert radians to degrees, use the conversion factor $\pi$ radians = 180 degrees. Consequently, $1$ radian = $(180/\pi)$ degrees.

For example, to convert $-\frac{2\pi}{3}$ to degrees, multiply by $(180/\pi)$, resulting in $-120$ degrees. This allows you to use a protractor or compass effectively to draw accurate angles.

headline of the respective core concept

Negative angles are measured clockwise from the positive x-axis, unlike positive angles, which are measured counterclockwise. This distinction is crucial in constructing and interpreting angles.

For instance, an angle of $\-\pi$ radians translates to $-180$ degrees, positioning the angle on the negative x-axis. Similarly, an angle of $-\frac{2\pi}{3}$ radians converts to $-120$ degrees when turning clockwise. Understanding negative angles helps in solving complex trigonometric problems and constructing precise geometric figures.

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91影视

Construct angles with the following radian measure. \(\pi / 6,-2 \pi / 3,-\pi\)

Short Answer

Step by step solution

Construct \(\frac{\pi}{6}\)

Construct \(-\frac{2\pi}{3}\)

Construct \(-\pi\)

Key Concepts

headline of the respective core concept

headline of the respective core concept

headline of the respective core concept

headline of the respective core concept

headline of the respective core concept

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