Question

The inclination of the line \(\sqrt{3} \mathrm{x}-\mathrm{y}+5=0\) with \(\mathrm{X}\) -axis is (1) \(90^{\circ}\) (2) \(45^{\circ}\) (3) \(60^{\circ}\) (4) \(30^{\circ}\)

Short answer

Answer: The inclination of the line with the x-axis is 60° (Option 3).

Step-by-step solution

Convert the equation to the form y = mx + c

Start by rewriting the given equation: √3x - y + 5 = 0 Add y and subtract 5 from both sides: y = √3x - 5 Now, the equation is in the form y = mx + c, with m = √3.

Calculate the slope

From the equation y = √3x - 5, we see that the slope is m = √3.

Find the angle θ using the tangent relationship

Now, we will use the formula tan(θ) = m to find the angle θ: tan(θ) = √3 θ = arctan(√3) We know that arctan(√3) corresponds to 60 degrees, so: θ = 60°

Check which multiple-choice answer corresponds to the calculated angle

We found that the angle θ is 60°. Comparing it with the available options, we see that it matches with option (3), therefore: The inclination of the line with the x-axis is 60° (Option 3).

Key concepts

Slope of a Line

When we talk about the slope of a line in mathematics, we are referring to a measure of the steepness or incline of that line. It's essentially the ratio of the vertical change, called the rise, to the horizontal change, called the run, between two points on a line.

The formula to find the slope, represented by the letter 'm', is given by \( m = \frac{\text{rise}}{\text{run}} \), which can also be interpreted as \( m = \frac{\Delta y}{\Delta x} \), where \( \Delta y \) and \( \Delta x \) are the differences in the y-coordinates and the x-coordinates of two points, respectively.
In the context of the incline of a line with respect to the x-axis, the slope is the tangent of the angle the line makes with the x-axis. If a line has a positive slope, it means that it inclines upwards as it moves from left to right. Conversely, a negative slope indicates that the line descends as it goes from left to right. A zero slope means the line is horizontal, and an undefined slope means the line is vertical.
For the given equation \( \sqrt{3} \mathrm{x}-\mathrm{y}+5=0 \), rewriting it in the form \( y = mx + c \) reveals the slope \( m = \sqrt{3} \). This tells us that for every unit you move horizontally along the x-axis, the line moves \sqrt{3} units vertically.

Trigonometric Ratios

Trigonometric ratios form the core of trigonometry and are used to relate the angles of a triangle to the ratios of its sides. The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). These ratios are properties of angles and are crucial when solving various geometry and physics problems.

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. So, for an angle θ, \( \tan(\theta) = \frac{\text{Opposite side}}{\text{Adjacent side}} \).
But these angles don't just apply to triangles – they're also key to understanding the incline of lines on a graph. When considering the inclination of a line to the x-axis, the tangent ratio expresses the slope of the line, which is the tangent of the angle the line creates with the x-axis. For the equation of a line in the form \( y = mx + c \) where 'm' is the slope, \( \tan(\theta) = m \), wherein the angle θ is the one formed between the line and the x-axis. This trigonometric link allows us to calculate the angle if we know the slope, and vice versa.
In our example, the slope \( m = \sqrt{3} \) corresponds to \( \tan(\theta) = \sqrt{3} \) and this relationship will guide us in finding the measure of the angle θ using the arctangent function.

Arctangent Function

The arctangent function, denoted as \( \arctan \text{ or } \tan^{-1} \), is one of the inverse trigonometric functions. While the tangent function gives you the ratio of the sides of a triangle given an angle, the arctangent function allows you to find the measure of the angle when you know the tangent ratio. It's important for converting slope values back into angles, which is especially useful in calculus and analytic geometry.

Let's consider our inclination problem: given that \( \tan(\theta) = m = \sqrt{3} \), we want to find \( \theta \) using the arctangent function. By inputting \( \sqrt{3} \) into the arctangent, it returns the angle in radians or degrees whose tangent is \( \sqrt{3} \). The arctangent of \( \sqrt{3} \) is known to correspond to an angle of \( 60^\circ \) or \( \frac{\pi}{3} \) radians, because \( \sqrt{3} \) is the tangent of the standard angle \( 60^\circ \) in trigonometry.
Finding the arctangent can be done using scientific calculators or appropriate math software. This process is a fundamental aspect of solving problems involving angles and slopes, demonstrating the interconnectedness of algebra with trigonometry. In the original exercise, by identifying the slope as \( \sqrt{3} \) and finding its arctangent, we ascertain the inclination of the line with the x-axis to be \( 60^\circ \)—a process that exemplifies the practical application of the arctangent function.