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Chapter 6: Problem 42

Find the angle \(\theta\) (in radians and degrees) between the lines. \(5 x+2 y=16\) \(3 x-5 y=-1\)

Short Answer

Expert verified
The angle \(饾渻\) (in both degrees and radians), obtained after following the above steps, is the final answer.

Step by step solution

01

Identifying the slopes from the equations

We first need to write our equations in the form of \(y = mx + c\) to identify the slopes. So the first equation \(5x + 2y = 16\) becomes \(y = -5/2x + 8\). Here, \(m1 = -5/2\). The second equation \(3x - 5y = -1\) becomes \(y = 3/5x + 1/5\). Here, \(m2 = 3/5\).
02

Substituting the slopes into the formula for the angle

Now we will substitute values of \(m1, m2\) into the formula \( tan(饾渻) = \((m1-m2) / (1+m1*m2)\). By doing this, we get \( tan(饾渻) = \((-5/2 - 3/5) / (1+(-5/2)*(3/5))\). After simplifying the above expression we will find \( tan(饾渻)\).
03

Calculating the angle

The value we obtained for \( tan(饾渻)\) is the tangent of the angle we are looking for. To find this angle, we take the arc-tangent of the value \( tan(饾渻)\). The result should be given in radians. To convert from radians to degrees, we will multiply by 180 and divide by \(\pi\). Thus, if we denote that value by \(饾渻\), it should be calculated as follows: \(饾渻 = atan(tan(饾渻)) * 180 / \pi\).
04

Final Answer

Finally, double check the answer is reasonable. The angle between two lines should be a value between 0 and 180 degrees.

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