/*! 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 20 Find the degree measure of the a... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 20

Find the degree measure of the angle with the given radian measure. $$-2$$

Short Answer

Expert verified
The angle is approximately \(-114.59\) degrees.

Step by step solution

01

Understanding the Conversion Formula

To convert an angle from radians to degrees, use the conversion formula: degrees = radians \( \times \frac{180}{\pi} \). This involves multiplying the radian measure by \( \frac{180}{\pi} \).
02

Substituting Values

Substitute the given radian measure into the conversion formula. Here, the radian measure is \(-2\), so calculate \(-2 \times \frac{180}{\pi}\).
03

Doing the Calculation

Perform the multiplication to convert the radians to degrees. Calculate \(-2 \times \frac{180}{3.14159}\) which approximately equals \(-114.59\).
04

Rounding

Typically, angles in degrees are rounded to one or two decimal places. Thus, round \(-114.59\) to \(-114.59\) degrees.

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Key Concepts

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

angle measurement
Angles are fundamental in geometry and trigonometry. They are a measure of rotation and can be expressed in two primary units: degrees and radians.
Understanding these units is key when working with angles in mathematics and its applications, such as physics and engineering.
  • Radians: A radian is a unit of angle measurement that relates an angle to the radius of a circle. It is defined such that a full circle has an angle of \(2\pi\) radians.
  • Degrees: A degree splits a full circle into 360 equal parts, so a circle contains 360掳.
By grasping how radians and degrees represent angles, you can seamlessly switch between these units during calculations.
They enable you to interpret angles with precision in various mathematical contexts.
degree measure
The degree measure is a popular way to describe angles in many contexts, especially in everyday applications. Degrees are used because they simplify the division of a circle The reasons for adopting degrees are:
  • Easy Division: Degrees divide a circle into 360 parts, which is handy for dividing tasks and visualizing angles.
  • Accessibility: Degrees are easier to understand for most people, making them useful in everyday tasks such as navigation and map reading.
  • Standardization: Many fields rely on degrees, providing a consistent language for discussing angles.
Despite their practicality, degrees are usually less convenient for mathematical calculations than radians, making radians preferred for calculus and trigonometry.
conversion formula
Understanding the radians to degrees conversion formula is essential when you need to interchange measurements between these two units. The formula is:\[\text{degrees} = \text{radians} \times \left(\frac{180}{\pi}\right)\]This formula allows you to calculate the equivalent degree measure from a given radian measure.
In the example provided, you have a radian measure of
  • First, substitute \(-2\) into the formula: \[-2 \times \left(\frac{180}{\pi}\right)\]
  • Then perform the calculation: \[-2 \times \frac{180}{3.14159} \approx -114.59\]
  • Lastly, you round the result to one or two decimal places to get \-114.59掳. This is how you convert radians to degrees, using this straightforward and essential formula.

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Most popular questions from this chapter

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$\sec \theta, \quad \sin \theta ; \quad \theta \text { in Quadrant } 1$$

The CN Tower in Toronto, Canada, is the tallest free-standing structure in North America. A woman on the observation deck, \(1150 \mathrm{ft}\) above the ground, wants to determine the distance between two landmarks on the ground below. She observes that the angle formed by the lines of sight to these two landmarks is \(43^{\circ} .\) She also observes that the angle between the vertical and the line of sight to one of the landmarks is \(62^{\circ}\) and to the other landmark is \(54^{\circ} .\) Find the distance between the two landmarks.

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The range \(R\) and height \(H\) of a shot put thrown with an initial velocity of \(v_{0}\) ft's at an angle \(\theta\) are given by $$\begin{array}{l} R=\frac{v_{0}^{2} \sin (2 \theta)}{g} \\ H=\frac{v_{0}^{2} \sin ^{2} \theta}{2 g} \end{array}$$ On the earth \(q=32 \mathrm{ft} / \mathrm{s}^{2}\) and on the moon \(g=5.2 \mathrm{ft} / \mathrm{s}^{2} .\) Find the range and height of a shot put thrown under the given conditions. (a) On the earth with \(v_{0}=12 \mathrm{ft} / \mathrm{s}\) and \(\theta=\pi / 6\) (b) On the moon with \(u_{b}=12\) ft \(/ s\) and \(\theta=\pi / 6\)

Find the quadrant in which \(\theta\) lies from the information given. $$\tan \theta<0 \quad \text { and } \quad \sin \theta<0$$
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