Problem 2
Question
If a nonvertical line has inclination \(\theta\) and slope \(m,\) then \(m=\)_____.
Step-by-Step Solution
Verified Answer
The slope \(m\) of a nonvertical line given by an inclination \(\theta\) is \(m = \tan(\theta)\).
1Step 1: Understand the Definition of Inclination and Slope
The inclination of a line, or the angle of inclination, is the angle from the positive x-axis to the line measured in the counterclockwise direction. It's usually represented by \(\theta\). The slope \(m\) of a line, on the other hand, is the 'steepness' or the 'gradient' of the line and it's given by the tangent of the angle of inclination. Thus we equate it as \(m = \tan(\theta)\).
2Step 2: Write the solution
The slope of a non-vertical line with an inclination of \(\theta\) can be computed with its tangent.
Key Concepts
Inclination of a lineAngle of inclinationTangent function
Inclination of a line
The inclination of a line is an important geometric concept. It refers to the angle formed between a line and the positive x-axis, measured in a counterclockwise direction. The angle of inclination helps us understand the line's orientation in a plane.
- **Positive Inclination:** When the line goes uphill from left to right, indicating an upward slope with the x-axis.
- **Zero Inclination:** This occurs when the line is perfectly horizontal and parallel to the x-axis.
- **Negative Inclination:** When the line goes downhill from left to right, showing a downward slope with the x-axis.
Understanding inclination is fundamental when dealing with lines in coordinate geometry. It provides insight into how steep or flat a line is and whether the line rises or falls as it moves from left to right.
- **Positive Inclination:** When the line goes uphill from left to right, indicating an upward slope with the x-axis.
- **Zero Inclination:** This occurs when the line is perfectly horizontal and parallel to the x-axis.
- **Negative Inclination:** When the line goes downhill from left to right, showing a downward slope with the x-axis.
Understanding inclination is fundamental when dealing with lines in coordinate geometry. It provides insight into how steep or flat a line is and whether the line rises or falls as it moves from left to right.
Angle of inclination
The angle of inclination is the specific angle a line makes with the positive x-axis. It tells us several things:
- **Direction of the Line:** By knowing the angle, we can tell if a line is going upwards or downwards.
- **Angular Measurement:** Usually represented by the symbol \(\theta\), this angle is measured in degrees or radians.
- **Impact on Line Slope:** The angle directly affects the slope of the line. For example, as the angle increases, the slope becomes steeper.
This concept is crucial because it allows mathematicians and scientists to describe the position and slope of lines precisely. The angle of inclination provides a bridge between linear algebra and trigonometry, linking geometric and algebraic concepts.
- **Direction of the Line:** By knowing the angle, we can tell if a line is going upwards or downwards.
- **Angular Measurement:** Usually represented by the symbol \(\theta\), this angle is measured in degrees or radians.
- **Impact on Line Slope:** The angle directly affects the slope of the line. For example, as the angle increases, the slope becomes steeper.
This concept is crucial because it allows mathematicians and scientists to describe the position and slope of lines precisely. The angle of inclination provides a bridge between linear algebra and trigonometry, linking geometric and algebraic concepts.
Tangent function
The tangent function is a key concept in trigonometry and plays a significant role in calculating the slope of a line. In the context of line inclination and slope:
- **Relationship with Inclination:** The slope \(m\) of a line is the tangent of its angle of inclination \(\theta\). It's denoted as \(m = \tan(\theta)\).
- **Tangent and Steepness:** A larger tangent value indicates a steeper line, while a smaller value represents a gentler slope.
- **Undefined Tangent:** When the line is vertical, the angle approaches \(90^\circ\) or \(\frac{\pi}{2}\) radians, and the tangent function becomes undefined.
Using the tangent function lets us convert angular measurements into practical slopes. This connection is invaluable when analyzing the direction and gradient of lines, making the tangent function integral in both geometry and trigonometry.
- **Relationship with Inclination:** The slope \(m\) of a line is the tangent of its angle of inclination \(\theta\). It's denoted as \(m = \tan(\theta)\).
- **Tangent and Steepness:** A larger tangent value indicates a steeper line, while a smaller value represents a gentler slope.
- **Undefined Tangent:** When the line is vertical, the angle approaches \(90^\circ\) or \(\frac{\pi}{2}\) radians, and the tangent function becomes undefined.
Using the tangent function lets us convert angular measurements into practical slopes. This connection is invaluable when analyzing the direction and gradient of lines, making the tangent function integral in both geometry and trigonometry.
Other exercises in this chapter
Problem 1
An __________ is the set of all points \((x, y)\) in a plane, the sum of whose distances from two distinct fixed points, called _________, is constant.
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Fill in the blanks. For the point \((r, \theta), r\) is the ___ ____ from \(O\) to \(P\) and \(\theta\) is the ____ _____ counterclockwise from the polar axis t
View solution Problem 2
Fill in the blanks. The _______ of a curve is the direction in which the curve is traced out for increasing values of the parameter.
View solution Problem 2
Fill in the blanks. The graph of a hyperbola has two disconnected parts called ________.
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