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Chapter 6: Problem 50

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{aligned} &5 x+2 y=16\\\ &3 x-5 y=-1 \end{aligned}$$

Short Answer

Expert verified
The angle \(胃\) between the two lines is approximately 1.19 radians or 68.2 degrees.

Step by step solution

01

Find the slope of the first line

Simplify the first equation to the form \(y = mx + b\). In this case, the slope of the first line (m1) can be calculated as \(-A/B = -5/2 = -2.5\).
02

Find the slope of the second line

Likewise, simplify the second equation to the form \(y = mx + b\). In this case, the slope of the second line (m2) can be calculated as \(-A/B = -3/5 = -0.6\).
03

Find the angle between the lines

Now, the angle between the lines can be calculated using the formula: \(tan(胃) = abs((b1 - b2) / (1 + b1 * b2))\). Substituting b1 = -2.5 and b2 = -0.6, the angle in radians is \(\theta_{rad} = arctan(abs((-0.6 - -2.5) / (1 + -0.6 * -2.5)))\).
04

Convert the angle from radians to degrees

Using the formula: \(胃_{deg} = 胃_{rad} * 180/蟺\), we find the angle in degrees.

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Most popular questions from this chapter

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$x^{2}+y^{2}=9 a^{2}$$

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