Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Coordinate Geometry → Straight Lines ↓


Angle Between Lines


In coordinate geometry, a line is a set of points that extends infinitely in both directions. When two lines intersect at a point, they form an angle. Studying the relationship between angles and lines is important for understanding the geometry of shapes and space. The concept of the angle between lines is a fundamental topic in coordinate geometry, and it provides valuable information about how lines relate to one another.

Let us first understand what it means when two lines intersect each other. Basically, two lines intersecting means that they share the same point. At this point, they form an angle, which can be measured to understand the relationship between these lines.

Understanding slope

Before diving into the angle between lines, it is necessary to understand the concept of slope. Slope measures the steepness or inclination of a line and is usually represented by the letter m. In a two-dimensional plane, the slope of a line through two points (x1, y1) and (x2, y2) is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

The slope m can be a positive or negative number:

  • Positive slope: The line increases from left to right.
  • Negative slope: The line falls from left to right.

Angle between two lines

When we talk about the angle between two straight lines, we are referring to the smallest angle through which one line can be rotated to align with the other line. This angle ranges between 0° and 180°. The mathematical relationship to find this angle involves the slopes of the lines.

Suppose we have two lines with slopes m1 and m2. The angle θ between two lines can be calculated using the formula:

tan(θ) = |(m2 - m1) / (1 + m1 * m2)|

Here:

  • tan(θ) is the tangent of the angle between two lines.
  • The vertical bars | ... | represent absolute value, and ensure that the tangent function returns a positive value.
  • The formula assumes that none of the lines are perpendicular. If one of the lines is perpendicular, the slope is undefined, and the angle will have to be calculated differently.

Special cases

Case 1: The lines are parallel:
If the lines are parallel, m1 = m2, and the formula gives:

tan(θ) = |(m2 - m1) / (1 + m1 * m2)| = 0

Which means θ = 0°. So parallel lines do not make any angle.

Case 2: The lines are perpendicular:
If the lines are perpendicular, then the product of their slopes is -1:

m1 * m2 = -1

In this case, the formula simplifies to:

tan(θ) = |(0) / (0)| = tan(90°)

Which means θ = 90°. Therefore, perpendicular lines intersect at right angles.

Example problems

Let's look at some examples to better understand calculating the angle between two lines.

Example 1: Find the angle between the lines whose equations are:

  • y = 2x + 3
  • y = (1/2)x - 1

First, find the slope of the given lines:

  • m1 = 2
  • m2 = 1/2

Use the formula for the angle between two lines:

tan(θ) = |(m2 - m1) / (1 + m1 * m2)|

Substitute m1 and m2:

tan(θ) = |((1/2) - 2) / (1 + (2)*(1/2))|

Simplification:

tan(θ) = |(-3/2) / (1 + 1)| = |-3/2| = 3/2

Calculate the angle using the inverse tangent function:

θ = tan-1 (3/2)

This will give you the angle in degrees or radians, depending on your calculator settings.

Visual explanations

θ y = 2x + 3 y = (1/2)x – 1

In this visual representation, the blue line represents the equation y = 2x + 3, and the red line represents y = (1/2)x - 1 The angle θ between the two lines is the angle at their intersection.

Understanding the angle between lines is essential for representing how objects relate to one another in space. This concept is particularly useful in fields such as engineering, architecture, and even in daily life scenarios such as understanding road intersections or the angles needed to cut materials.

The methods discussed here provide a straightforward way to calculate and visualize the angle between two lines when their equations are known. By understanding and applying these mathematical concepts, we can enhance our understanding of geometry and improve our problem-solving skills.


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Angle Between Lines

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