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Chapter 6: Problem 52

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{aligned} &3 x-5 y=3\\\ &3 x+5 y=12 \end{aligned}$$

Short Answer

Expert verified
The angle between the two lines is -1.11 radians or -63.43 degrees.

Step by step solution

01

Find the slopes

Rewrite the two equations in the form \(y = mx + c\), where m indicates the slope of each function. The first equation \(-3x+5y=3\) becomes \(y=\frac{3}{5}x + \frac{3}{5}\), so the slope \(m1=\frac{3}{5}\). The second equation \(3x+5y=12\) becomes \(y = -\frac{3}{5}x + \frac{12}{5}\), so the slope \(m2 = -\frac{3}{5}\).
02

Calculate the angle

Use the slopes of the lines in the formula for \(\theta = \arctan \left( \frac{m2-m1}{1+m1 \cdot m2} \right)\). Substituting in the values, we get \(\theta = \arctan \left( \frac{-\frac{3}{5}-\frac{3}{5}}{1+ \frac{3}{5} \cdot -\frac{3}{5}} \right)\). This simplifies to \(\theta = \arctan(-2)\). Taking the arctan of -2 in radians gives us \(-1.11\) radians.
03

Convert radians to degrees

We have the angle in radians but we also need it in degrees. The formula for converting radians to degrees is \(degrees = radians \times \frac{180}{\pi}\). Substituting our radian value into that equation gives us \(-63.43\) degrees.

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Most popular questions from this chapter

Conjecture Consider the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, \quad a+b=20\) (a) The area of the ellipse is given by \(A=\pi a b .\) Write the area of the ellipse as a function of \(a .\) (b) Find the equation of an ellipse with an area of 264 square centimeters. (c) Complete the table using your equation from part (a). Then make a conjecture about the shape of the ellipse with maximum area. $$\begin{array}{|l|l|l|l|l|l|l|} \hline a & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline A & & & & & & \\ \hline \end{array}$$ (d) Use a graphing utility to graph the area function and use the graph to support your conjecture in part (c).

Convert the polar equation $$r=2(h \cos \theta+k \sin \theta)$$ to rectangular form and verify that it is the equation of a circle. Find the radius of the circle and the rectangular coordinates of the center of the circle.

Find the slope of the line with inclination \(\boldsymbol{\theta}\) $$\theta=1.27 \text { radians }$$

In Exercises \(5-18\), plot the point given in polar coordinates and find two additional polar representations of the point, using \(-2 \pi < \theta < 2 \pi\). $$(2 \sqrt{2}, 4.71)$$

Sketching an Ellipse In Exercises \(33-48\), find the center, vertices, foci, and eccentricity of the ellipse. Then sketch the ellipse. $$\frac{(x+5)^{2}}{9 / 4}+(y-1)^{2}=1$$
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