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Chapter 4: Problem 42

Convert the angle measure from degrees to radians. Round to three decimal places. $$345^{\circ}$$

Short Answer

Expert verified
After performing the multiplication and rounding to three decimal places, the result is \( 6.021 \) radians. This is the equivalent of \( 345^{\circ} \) in radians.

Step by step solution

01

Identify the given degree measurement

The given angle measurement is \(345^{\circ}\).
02

Apply the conversion factor

To convert the degree measure to radians, multiply it by the conversion factor \( \frac{\pi}{180} \). So the calculation becomes \( 345^{\circ} \times \frac{\pi}{180} \).
03

Perform the calculation

Carry out the multiplication to find the radian measure. It is necessary to use a calculator that can handle the value of pi for this operation. Make sure to round the answer to three decimal places as requested.

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Key Concepts

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

Angle Measurement
Angles are a fundamental aspect of geometry and trigonometry. They help us describe rotations and define how two lines meet at a point. They are typically measured in either degrees or radians. Degrees are the more common units found in everyday use. They divide a circle into 360 equal parts. So, a right angle is 90 degrees. On the other hand, radians are units used more in advanced mathematics and physics.

When dealing with degrees, it鈥檚 crucial to understand that:
  • A full circle is 360 degrees
  • A half-circle, or straight angle, is 180 degrees
  • A quarter circle, or right angle, is 90 degrees
To work effectively with angles, especially in higher-level mathematics, being comfortable with both degrees and radians and converting between them is essential.
Conversion Factor
The conversion factor is what allows us to switch between measuring angles in degrees and radians. A conversion factor is essentially a multiplier that lets you change from one unit to another. For angles, this factor is necessary because degrees and radians measure the same thing but in different ways.

The key relationship between degrees and radians is governed by the circle's properties:
  • The whole circle is 360 degrees, or it can also be described as \(2\pi\) radians
  • Thus, 180 degrees is equivalent to \(\pi\) radians
  • To convert from degrees to radians, use the factor \(\frac{\pi}{180}\)
  • To convert from radians to degrees, use the factor \(\frac{180}{\pi}\)
By remembering these factors, you can easily switch back and forth between these systems of measurement.
Radian Measure
Radian measure is a way of expressing angles using the radius of a circle. In the radian system, an angle is defined by the length of the arc that the angle subtends on a unit circle. This means that for a circle with a radius of one, an angle's measure in radians is simply the length of the related arc.

Here are some important points:
  • One radian is the angle made by taking the radius of a circle and wrapping it along the circle's edge
  • A complete circle has an arc length equal to the circumference, \(2\pi\) times the radius, so a full circle equals \(2\pi\) radians
  • Radians provide a direct and natural measure of angles that are particularly useful in calculus, as they lead to simpler derivatives and integrals when dealing with trigonometric functions
Understanding radians can initially be a challenge, especially if you are used to thinking in degrees. However, becoming familiar with this measure will greatly assist you in higher mathematics, where radians are the standard unit.

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Most popular questions from this chapter

Graph \(f\) and \(g\) in the same coordinate plane. Include two full periods. Make a conjecture about the functions. $$f(x)=\sin x, \quad g(x)=-\cos \left(x+\frac{\pi}{2}\right)$$

Consider the function \(f(x)=x-\cos x\) (a) Use a graphing utility to graph the function and verify that there exists a zero between 0 and 1 . Use the graph to approximate the zero. (b) Starting with \(x_{0}=1,\) generate a scquence \(x_{1}, x_{2}\) \(x_{3}, \ldots,\) where \(x_{n}=\cos \left(x_{n-1}\right) .\) For example, \(x_{0}=1\) \(x_{1}=\cos \left(x_{0}\right)\) \(x_{2}=\cos \left(x_{1}\right)\) \(x_{3}=\cos \left(x_{2}\right)\) \(\vdots\) What value does the sequence approach?

Find two solutions of each equation. Give your answers in degrees \(\left(0^{\circ} \leq \theta<360^{\circ}\right)\) and in radians \((0 \leq \theta<2 \pi) .\) Do not use a calculator. (a) \(\cos \theta=\frac{\sqrt{2}}{2}\) (b) \(\cos \theta=-\frac{\sqrt{2}}{2}\)

The numbers of hours \(H\) of daylight in Denver, Colorado, on the 15 th of each month are: \(1(9.67), 2(10.72), 3(11.92), 4(13.25)\) \(5(14.37), \quad 6(14.97), \quad 7(14.72), \quad 8(13.77), \quad 9(12.48)\) \(10(11.18), \quad 11(10.00), \quad 12(9.38) . \quad\) The month is represented by \(t,\) with \(t=1\) corresponding to January. A model for the data is $$H(t)=12.13+2.77 \sin \left(\frac{\pi t}{6}-1.60\right)$$. (a) Use a graphing utility to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the context of the problem? Explain.

After exercising for a few minutes, a person has a respiratory cycle for which the velocity of airflow is approximated by $$v=1.75 \sin \frac{\pi t}{2}$$ where \(t\) is the time (in seconds). (Inhalation occurs when \(v > 0,\) and exhalation occurs when \(v < 0 .\) ) (a) Find the time for one full respiratory cycle. (b) Find the number of cycles per minute. (c) Sketch the graph of the velocity function.
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