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Chapter 10: Problem 44

Find the inclination \(\theta\) (in radians and degrees) of the line. \(5 x+3 y=0\)

Short Answer

Expert verified
The inclination of the line is \(\theta = \arctan(-5/3)\) in radians and \(\theta_{\text{degrees}} = \theta_{\text{radians}} * (180/\pi)\) in degrees.

Step by step solution

01

Identify coefficients

The coefficients can be gotten from the given equation. Here, \(A = 5\) and \(B = 3\).
02

Calculate inclination in radians

Substitute the coefficients \(A\) and \(B\) into the formula \(\theta = \arctan(-A/B)\) to find the inclination in radians. So, the inclination \(\theta\) in radians is \(\theta = \arctan(-5/3) \).
03

Convert radians to degrees

We then convert the radians to degrees using the conversion factor \(180/\pi\). So, to convert the inclination \(\theta\) found in step 2 to degrees, we use the formula \(\theta_{\text{degrees}} = \theta_{\text{radians}} * (180/\pi)\).

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Key Concepts

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

Radians
Radians are a fundamental unit of angular measurement. Unlike degrees, which divide a circle into 360 parts, radians are defined by the arc length of a circle.
One radian is the angle created when the arc length is equal to the radius of the circle.
This makes radians a natural measure in mathematics, especially in calculus and trigonometry.
  • There are approximately 6.283 radians in a full circle. This is because the circumference of a circle is given by the formula \(2\pi r\), where \(r\) is the radius.
  • Thus, a full circle is \(2\pi\) radians.
  • Since the radian measures actual lengths along the circumference, it provides a direct relationship with the geometry of the circle.
Understanding radians is crucial for correctly interpreting angles in many areas of mathematics.
Degrees
Degrees are perhaps the most familiar unit of measuring angles. A complete circle is divided into 360 degrees in this system.
This division makes degrees a convenient unit for everyday use and simple geometric calculations.
  • Each degree is subdivided into 60 minutes, and each minute is further divided into 60 seconds.
  • Historically, the 360-degree system is believed to originate from the ancient Babylonians, who used a base 60 number system.
  • Degrees are typically used in many practical applications, such as navigation, engineering, and construction, because of their simplicity.
Despite being less precise than radians in higher mathematical calculations, degrees are an accessible way to describe angles.
Arctangent
The arctangent function, denoted as \( \arctan \) or \( \tan^{-1} \), is used to find an angle whose tangent is a specific value. For instance, in this exercise, we used \( \arctan(-5/3) \) to find the angle of inclination.
  • The arctangent function is essentially the inverse of the tangent function.
  • It takes a ratio of opposite to adjacent sides in a right triangle and returns the corresponding angle in radians.
  • The result is generally valued between \(-\pi/2\) and \(\pi/2\) radians.
In solving linear equations, arctangent helps connect ratios of coefficients to actual angles, making it a powerful tool for determining inclinations.
Angle Conversion
Converting between radians and degrees is a common task when working with angles. Each system has its merits, so understanding how to switch between them is useful.
The key relationship is that \( \pi \) radians is equal to 180 degrees.
  • To convert from radians to degrees, use the formula: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \).
  • Conversely, to convert from degrees to radians, use: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).
  • These conversions are important in contexts where calculations are done in radians, but outcomes need to be interpreted in more familiar units of degrees, such as in navigation or certain engineering fields.
Knowing how to convert angles ensures flexibility and precision in both theoretical and practical applications.

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Most popular questions from this chapter

In Exercises \(23-48\) , sketch the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $$r=4 \cos \theta$$

In Exercises 65-68, use a graphing utility to graph the polar equation and show that the indicated line is an asymptote of the graph. $$\begin{array}{ll}{\text { Name of Graph }} & {\text { Polar Equation }} & {\text { Asymptote }} \\ {\text { conchoid }} & {r=2+\csc \theta} & {y=1}\end{array}$$

Converting a Polar Equation to Rectangular Form In Exercises \(117-126,\) convert the polar equation to rectangular form. Then sketch its graph. $$r=2 \sin \theta$$

Converting a Polar Equation to Rectangular Form In Exercises \(91-116,\) convert the polar equation to rectangular form. $$r=\frac{6}{2-3 \sin \theta}$$

In Exercises \(23-48\) , sketch the graph of the polar equation using symmetry, zeros, maximum r-values, and any other additional points. $$r=-7$$
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