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Chapter 6: Problem 44

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

Short Answer

Expert verified
The angle of inclination \(\theta\) of the line \(5x + 3y = 0\) is \( \theta = arctan(-5/3)\) radians and \( \theta_{degrees} = \theta_{radians} \times (180/\pi)\) degrees.

Step by step solution

01

Find the Slope of the Line

The slope of the line defined by \(5x+3y=0\) is \(-a/b\). Substituting the coefficients from the equation into this formula, we get the slope \(m = -a/b = -5/3\).
02

Calculate the Angle of Inclination in Radians

Knowing that the tangent of the inclination angle equals the slope of the line, you can write the equation \(tan(\theta) = m.\) Substituting the value found for the slope into this equation, \(tan(\theta) = -5/3\), you can solve for the angle of inclination in radians by calculating the arctangent of the slope. So \(\theta = arctan(-5/3)\).
03

Convert to Degrees

To convert the angle from radians to degrees, use the formula \(degrees = radians \times (180/\pi)\). So, the inclination angle in degrees is given by \( \theta_{degrees} = \theta_{radians} \times (180/\pi)\).

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Most popular questions from this chapter

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$y^{2}=x^{3}$$

The equation \(r=\frac{e p}{1 \pm e \sin \theta}\) is the equation of an ellipse with \(e<1 .\) What happens to the lengths of both the major axis and the minor axis when the value of \(e\) remains fixed and the value of \(p\) changes? Use an example to explain your reasoning.

The points represent the vertices of a triangle. (a) Draw triangle \(A B C\) in the coordinate plane, (b) find the altitude from vertex \(B\) of the triangle to side \(A C,\) and \((\mathrm{c})\) find the area of the triangle. $$A(-2,0), B(0,-3), C(5,1)$$

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$\theta=3 \pi / 4$$

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=-2 \cos \theta$$
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