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Chapter 5: Problem 41

What angle does the line \(y=\frac{2}{5} x\) in the \(x y\) -plane make with the positive \(x\) -axis?

Short Answer

Expert verified
The line \(y = \frac{2}{5}x\) makes an angle of approximately \(21.80掳\) with the positive x-axis.

Step by step solution

01

Determine the slope of the line

For a line with the equation \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y-intercept, we can easily determine the slope from the given equation \(y = \frac{2}{5}x\). The slope of the line, \(m\), is equal to \(\frac{2}{5}\).
02

Use the tangent function to find the angle

The tangent function relates the angle of a right triangle to the ratio of its opposite and adjacent sides. In our case, the angle we are looking for is the angle between the line and the positive x-axis. Now we know the slope of the line is \(\frac{2}{5}\), and we can think of that as the ratio of the vertical change (rise) to the horizontal change (run) in the line. So for every 2 units the line goes up, it moves 5 units to the right. In a right triangle, we can say that the opposite side is 2 and the adjacent side is 5. Since the tangent function is defined as \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\), we have: \(\tan(\theta) = \frac{2}{5}\)
03

Find the angle using the arctangent function

Now we want to find the angle \(\theta\) such that \(\tan(\theta) = \frac{2}{5}\). To do so, we will use the arctangent function, also known as the inverse tangent function. \(\theta = \arctan\left(\frac{2}{5}\right)\) Using a calculator, we get: \(\theta \approx 21.80掳\) So, the angle that the line \(y=\frac{2}{5}x\) makes with the positive x-axis is approximately 21.80掳.

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Most popular questions from this chapter

The next two exercises emphasize that \(\cos (x+y)\) does not equal \(\cos x+\cos y\). For \(x=19^{\circ}\) and \(y=13^{\circ}\), evaluate each of the following: (a) \(\cos (x+y)\) (b) \(\cos x+\cos y\)

Find a formula for \(\cos \left(\theta+\frac{\pi}{4}\right)\).

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