Problem 37
Question
Find the inclination \(\theta\) (in radians and degrees) of the line. $$3 x-3 y+1=0$$
Step-by-Step Solution
Verified Answer
The inclination \(\theta\) of the given line is \(\frac{\pi }{4}\) radians or 45 degrees.
1Step 1: Identify Coefficient a and b from given equation
From the given equation \(3x - 3y + 1 = 0\), here coefficient of \(x (a)\) is 3 and coefficient of \(y (b)\) is -3.
2Step 2: Calculate the inclination in radians
Inclination can be computed as \(\theta = \arctan(\frac{-a}{b})\), this gives \(\theta = \arctan(\frac{-3}{-3}) = \arctan(1)\). Due to the standard inverse tangent values, \(\theta = \arctan(1) =\frac{\pi }{4}\).
3Step 3: Convert the inclination to degrees
To convert the value of \(\theta\) from radians to degrees, use the relation that 180 degrees is equal to \(\pi\) radians. Therefore, \(\theta = \frac{\pi }{4}(\frac{180}{\pi}) = 45\) degrees.
Key Concepts
Slope and InclinationConverting Radians to DegreesInverse Trigonometric Functions
Slope and Inclination
Understanding the slope and inclination of a line is a fundamental aspect of algebra and geometry that enables one to decipher the direction and steepness of a line. The slope of a line in a two-dimensional plane describes how much the line rises (or falls) vertically for each unit of horizontal change. Graphically, if you imagine you're walking along the line, the slope tells you how steep the climb (or descent) is.
The inclination of a line, on the other hand, is an angle, usually denoted as \(\theta\), that the line makes with the positive direction of the x-axis. This measurement is taken counterclockwise from the x-axis to the line itself. It's an alternative way of describing the slope since knowing one can determine the other.
The slope could be positive, negative, zero, or undefined, which corresponds to upward sloping lines, downward sloping lines, horizontal lines, and vertical lines, respectively. For instance, in a simple line equation of the form \(y = mx + b\), \(m\) represents the slope. But in terms of inclination, the same slope is associated with the tangent of the angle of inclination, i.e., \(m = \tan(\theta)\). Therefore, for the given exercise, where the line's equation is \(3x - 3y + 1 = 0\), the slope can be understood as \(\tan(\theta) = 1\), which indicates the line rises one unit vertically for each horizontal unit and has an inclination of \(\theta = \arctan(1)\).
The inclination of a line, on the other hand, is an angle, usually denoted as \(\theta\), that the line makes with the positive direction of the x-axis. This measurement is taken counterclockwise from the x-axis to the line itself. It's an alternative way of describing the slope since knowing one can determine the other.
The slope could be positive, negative, zero, or undefined, which corresponds to upward sloping lines, downward sloping lines, horizontal lines, and vertical lines, respectively. For instance, in a simple line equation of the form \(y = mx + b\), \(m\) represents the slope. But in terms of inclination, the same slope is associated with the tangent of the angle of inclination, i.e., \(m = \tan(\theta)\). Therefore, for the given exercise, where the line's equation is \(3x - 3y + 1 = 0\), the slope can be understood as \(\tan(\theta) = 1\), which indicates the line rises one unit vertically for each horizontal unit and has an inclination of \(\theta = \arctan(1)\).
Converting Radians to Degrees
Radians and degrees are two units for measuring angles, and they can be converted from one to the other using a simple ratio. Since there are \(2\pi\) radians in a full circle, and a full circle is 360 degrees, therefore \(\pi\) radians is equivalent to 180 degrees. The conversion factor between radians and degrees is based on this relationship: \(1\text{ radian} = \frac{180}{\pi}\text{ degrees}\).
To convert an angle in radians to degrees, one can multiply the radian measure by the conversion factor \(\frac{180}{\pi}\). For instance, if you are given the angle \(\theta = \frac{\pi}{4}\) radians and wish to express it in degrees, you would calculate as follows: \(\theta_{degrees} = \theta_{radians} \times \frac{180}{\pi} = \frac{\pi}{4} \times \frac{180}{\pi} = 45^\text{\textdegree}\). Here, the \(\pi\)'s cancel out, leaving you with a clean conversion to 45 degrees, just as demonstrated in the step by step solution.
To convert an angle in radians to degrees, one can multiply the radian measure by the conversion factor \(\frac{180}{\pi}\). For instance, if you are given the angle \(\theta = \frac{\pi}{4}\) radians and wish to express it in degrees, you would calculate as follows: \(\theta_{degrees} = \theta_{radians} \times \frac{180}{\pi} = \frac{\pi}{4} \times \frac{180}{\pi} = 45^\text{\textdegree}\). Here, the \(\pi\)'s cancel out, leaving you with a clean conversion to 45 degrees, just as demonstrated in the step by step solution.
Inverse Trigonometric Functions
Inverse trigonometric functions are used to find angles when the trigonometric ratios (sine, cosine, tangent, etc.) are known. They are essentially the reverse of the trigonometric functions, answering the question: Given a ratio, what is the angle that produced this ratio?
The most commonly used inverse trigonometric functions include \(\arcsin\), \(\arccos\), and \(\arctan\), corresponding to the functions sine, cosine, and tangent, respectively. These are often represented as \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\), but this notation can be confused with reciprocal functions, so \(\arcsin\), \(\arccos\), and \(\arctan\) are preferred in many contexts.
When you employ an inverse trigonometric function, such as \(\arctan(\frac{-a}{b})\) in the provided solution, you're seeking the angle \(\theta\) whose tangent is \(\frac{-a}{b}\). If both \(a\) and \(b\) are negative, as in our equation, the negatives cancel out, indicating the line is rising to the right, and we find a first quadrant angle, which is a standard position angle. Therefore, in the solution when we find \(\theta = \arctan(1)\), we identify the angle whose tangent is 1. According to the unit circle where trigonometric functions are defined, this angle is \(\frac{\pi}{4}\) radians or 45 degrees, illuminating the relationship between the line's inclination and the angle found via the inverse trigonometric function.
The most commonly used inverse trigonometric functions include \(\arcsin\), \(\arccos\), and \(\arctan\), corresponding to the functions sine, cosine, and tangent, respectively. These are often represented as \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\), but this notation can be confused with reciprocal functions, so \(\arcsin\), \(\arccos\), and \(\arctan\) are preferred in many contexts.
When you employ an inverse trigonometric function, such as \(\arctan(\frac{-a}{b})\) in the provided solution, you're seeking the angle \(\theta\) whose tangent is \(\frac{-a}{b}\). If both \(a\) and \(b\) are negative, as in our equation, the negatives cancel out, indicating the line is rising to the right, and we find a first quadrant angle, which is a standard position angle. Therefore, in the solution when we find \(\theta = \arctan(1)\), we identify the angle whose tangent is 1. According to the unit circle where trigonometric functions are defined, this angle is \(\frac{\pi}{4}\) radians or 45 degrees, illuminating the relationship between the line's inclination and the angle found via the inverse trigonometric function.
Other exercises in this chapter
Problem 37
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