/*! 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 31 Convert each radian measure to d... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 31

Convert each radian measure to degrees. $$\frac{7 \pi}{4}$$

Short Answer

Expert verified
315 degrees

Step by step solution

01

- Understand the relationship

Recall that to convert from radians to degrees, use the conversion factor \(\frac{180^\text{o}}{\text{蟺}}\).
02

- Set up the conversion

Multiply the given radian measure by \(\frac{180^\text{o}}{\text{蟺}}\). So, \(\frac{7\text{蟺}}{4} \times \frac{180^\text{o}}{\text{蟺}}\).
03

- Simplify

Cancel out \( \text{蟺} \) in the numerator and the denominator to get \( \frac{7 \times 180^\text{o}}{4} \), which simplifies to \( \frac{1260^\text{o}}{4} \).
04

- Final computation

Perform the division: \( \frac{1260^\text{o}}{4} = 315^\text{o} \).

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Key Concepts

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

radian measure
Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians divide a circle using the radius length. This means that in one full circle, there are 2蟺 radians. Radians provide a natural way of describing angles where the angle is the arc length divided by the radius of the circle. For example, an angle that cuts an arc equal to the circle's radius corresponds to 1 radian.
degree measure
Degrees are a more familiar way of measuring angles, especially in everyday use. An entire circle is divided into 360 degrees. This common system dates back to the ancient Babylonians, who preferred the number 360 for its divisibility properties. When working with degrees, it's simple to envision right angles (90掳) or straight lines (180掳). Degrees are particularly useful for navigation and in fields such as geometry and trigonometry.
conversion factor
The conversion factor between radians and degrees is crucial for switching from one unit to the other. This factor is based on the fact that a full circle is both 360掳 and 2蟺 radians. Therefore, the conversion factor from radians to degrees is \(\frac{180^\text{o}}{\text{蟺}} \). When converting, multiply the radian measure by this factor to get the degree measure. For example, to convert \(\frac{7 \text{蟺}}{4} \) to degrees, you multiply by \(\frac{180^\text{o}}{\text{蟺}} \). This results in \(\frac{7 \times 180^\text{o}}{4} = 315^\text{o} \). The factor helps streamline the process of converting between these two systems and ensures calculations remain accurate.

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Most popular questions from this chapter

Find the exact values of 6 in the given interval that satisfy the given condition. $$[-2 \pi, \pi) ; \quad 3 \tan ^{2} s=1$$

Graph each function over a two-period interval. $$y=\sin \left(x-\frac{\pi}{4}\right)$$

Find the exact value of \(s\) in the given interval that has the given circular function value. Do not use a calculator. $$\left[\pi, \frac{3 \pi}{2}\right] ; \quad \sin s=-\frac{1}{2}$$

Solve each problem. The voltage \(E\) in an electrical circuit is modeled by $$ E=5 \cos 120 \pi t $$ where \(t\) is time measured in seconds. (a) Find the amplitude and the period. (b) How many cycles are completed in 1 sec? (The number of cycles, or periods, completed in 1 sec is the frequency of the function.) (c) Find \(E\) when \(t=0,0.03,0.06,0.09,0.12\). (d) Graph \(E\) for \(0 \leq t \leq \frac{1}{30}\).

Solve each problem. The solar constant \(S\) is the amount of energy per unit area that reaches Earth's atmosphere from the sun. It is equal to 1367 watts per \(m^{2}\) but varies slightly throughout the seasons. This fluctuation \(\Delta S\) in \(S\) can be calculated using the formula $$ \Delta S=0.034 S \sin \left[\frac{2 \pi(82.5-N)}{365.25}\right] $$ In this formula, \(N\) is the day number covering a four-year period, where \(N=1\) corresponds to January 1 of a leap year and \(N=1461\) corresponds to December 31 of the fourth year. (Source: Winter, C., R. Sizmann, and L. L. Vant-Hull, Editors, Solar Power Plants, Springer-Verlag. (a) Calculate \(\Delta S\) for \(N=80\), which is the spring equinox in the first year. (b) Calculate \(\Delta S\) for \(N=1268\), which is the summer solstice in the fourth year. (c) What is the maximum value of \(\Delta S ?\) (d) Find a value for \(N\) where \(\Delta S\) is equal to \(0 .\)
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