Exhaustive Study Guide: Linear Relationships, Pythagoras, and Trigonometry

Linear Relationships

• Interpreting Straight-Line Graphs (Consolidating)

  • A straight-line graph represents a linear relationship between two variables, traditionally defined on a Cartesian plane with a horizontal axis ($x$-axis) and a vertical axis ($y$-axis).

  • The independent variable is plotted on the horizontal axis ($x$), while the dependent variable is plotted on the vertical axis ($y$).

  • Every point on a line corresponds to an ordered pair $(x, y)$, representing a solution to the underlying linear equation.

  • Key features of straight-line graphs include:

    • Direction: A line that rises from left to right exhibits a positive trend; a line that falls from left to right exhibits a negative trend.

    • Steepness: The inclination of the line indicates the rate at which $y$ changes relative to $x$.

    • Axis intersections: Points where the line crosses the coordinate axes are designated as intercepts.

• Distance-Time Graphs

  • A distance-time graph illustrates the movement of an object over a period of time, plotting distance on the vertical axis ($y$) against time on the horizontal axis ($x$).

  • The gradient (slope) of a distance-time graph represents the speed or velocity of the object:         $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

  • Interpretation of specific graph segments:

    • Horizontal line ($\text{gradient} = 0$): The object is stationary (at rest); distance does not change as time passes.

    • Constant positive slope: The object is moving away from the starting position at a constant speed.

    • Steeper line: Indicates a higher speed (greater distance covered per unit time).

    • Gentler line: Indicates a lower speed.

    • Constant negative slope: The object is returning toward the reference point or starting position at a constant speed.

    • Curved line: Represents changing speed (acceleration or deceleration).

  • Distinction between total distance and displacement:

    • Total distance is the cumulative length of all movement legs.

    • Displacement is the net straight-line distance from the initial starting position to the final position.

• Graphing Straight Lines (Consolidating)

  • Linear equations describe constant rates of change and can be graphed using multiple techniques:

  • Table of Values Method:

    1. Select a set of representative $x$-values (typically including negative numbers, zero, and positive numbers, such as $-2, -1, 0, 1, 2$).

    2. Substitute each $x$-value into the given linear equation to calculate the corresponding $y$-value.

    3. Form coordinate pairs $(x, y)$.

    4. Plot the points on a Cartesian grid.

    5. Draw a continuous straight line through all plotted points, extending past them with arrowheads to indicate infinite extension.

• Exploring Gradient

  • The gradient ($m$) of a non-vertical line is a numerical measure of its steepness and direction.

  • Mathematical definition of gradient:         $m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$         where $(x_1, y_1)$ and $(x_2, y_2)$ are any two distinct points on the straight line.

  • Four distinct orientation types for gradients:

    • Positive gradient ($m > 0$): The line rises from left to right ($y$ increases as $x$ increases).

    • Negative gradient ($m < 0$): The line falls from left to right ($y$ decreases as $x$ increases).

    • Zero gradient ($m = 0$): Horizontal line ($y$ remains constant regardless of $x$; $\text{rise} = 0$).

    • Undefined gradient ($m$ is undefined): Vertical line ($x$ remains constant regardless of $y$; $\text{run} = 0$, causing division by zero).

• Rates from Graphs

  • The gradient of a line on a contextual graph represents the rate of change of the vertical quantity with respect to the horizontal quantity.

  • Units of rate are calculated as:         $\text{Unit of Rate} = \frac{\text{Unit of Vertical Variable}}{\text{Unit of Horizontal Variable}}$

  • Examples of physical rates derived from linear gradients:

    • Distance vs. Time graph: Rate is speed, measured in $m\,s^{-1}$ or $km\,h^{-1}$.

    • Volume vs. Time graph: Rate is flow rate, measured in $L\,min^{-1}$ or $cm^3\,s^{-1}$.

    • Cost vs. Quantity graph: Rate is unit price, measured in $\text{dollars}\,kg^{-1}$ or $\text{dollars}\,\text{item}^{-1}$.

  • Average rate of change over an interval on a non-linear graph is found by calculating the gradient of the secant line joining the interval's endpoints.

• $y = mx + c$ and Special Lines

  • Gradient-Intercept Form:         $y = mx + c$         where $m$ represents the numerical gradient (slope) of the line, and $c$ represents the $y$-intercept value (the point $(0, c)$).

  • Special Lines:

    • Horizontal Lines:

      • Equation form: $y = c$ (where $c$ is a constant real number).

      • Gradient: $m = 0$.

      • Every point on the line shares the same $y$-coordinate.

    • Vertical Lines:

      • Equation form: $x = k$ (where $k$ is a constant real number).

      • Gradient: Undefined (vertical inclination).

      • Every point on the line shares the same $x$-coordinate.

      • Does not represent a mathematical function of $x$.

    • Direct Proportion Lines through the Origin:

      • Equation form: $y = mx$ (where $c = 0$).

      • Passes directly through the origin $(0, 0)$.

• Parallel Lines and Perpendicular Lines

  • Parallel Lines:

    • Lines that lie in the same plane and never intersect, maintaining a constant distance apart.

    • Condition for parallelism: Two non-vertical lines with gradients $m_1$ and $m_2$ are parallel if and only if their gradients are equal:             $m_1 = m_2$

    • Parallel lines have different $y$-intercepts ($c_1 \neq c_2$).

  • Perpendicular Lines:

    • Lines that intersect at a right angle ($90^\circ$).

    • Condition for perpendicularity: Two non-vertical lines with gradients $m_1$ and $m_2$ are perpendicular if and only if the product of their gradients is $-1$:             $m_1 \times m_2 = -1$

    • Equivalently, one gradient is the negative reciprocal of the other:             $m_2 = -\frac{1}{m_1}$

    • A horizontal line ($m = 0$) and a vertical line (undefined gradient) are always perpendicular.

• Graphing Straight Lines Using Intercepts (Extending)

  • The intercept method is an efficient technique for graphing linear equations written in general form or standard form ($Ax + By = C$).

  • Definitions:

    • $x$–intercept: The point where the line intersects the $x$-axis. At this point, $y = 0$. Expressed as $(x, 0)$.

    • $y$–intercept: The point where the line intersects the $y$-axis. At this point, $x = 0$. Expressed as $(0, y)$.

  • Procedure for graphing using intercepts:

    1. Find the $x$–intercept by setting $y = 0$ in the equation and solving for $x$.

    2. Find the $y$–intercept by setting $x = 0$ in the equation and solving for $y$.

    3. Plot the resulting points $(x, 0)$ and $(0, y)$ on the coordinate axes.

    4. Draw a continuous straight line extending through both intercept points.

  • Exception: If the line passes through the origin $(0, 0)$, both intercepts are at the same point. An additional point must be calculated using substitution to define the line's direction.

• Linear Modelling

  • Linear modelling uses linear equations to represent real-world quantitative situations where a variable changes at a constant rate.

  • General form of a linear model:         $C(x) = mx + k$         where $m$ represents the marginal rate of change (variable cost per unit), $x$ represents the quantity or domain input, and $k$ represents the fixed base value (initial cost or constant offset).

  • Data evaluation concepts:

    • Interpolation: Estimating an unknown value within the range of known data points. Interpolated estimates are generally reliable.

    • Extrapolation: Estimating an unknown value outside the range of known data points. Extrapolated values carry greater uncertainty as linear trends may not persist indefinitely.

• Direct Variation

  • Direct variation describes a simple mathematical relationship between two variables where one is a constant multiple of the other.

  • Algebraic formulation:         $y = kx$         where $k$ is the non-zero constant of variation (or constant of proportionality).

  • Calculating constant of variation:         $k = \frac{y}{x}$

  • Graphical properties:

    • The graph of a direct variation relationship is always a straight line passing through the origin $(0, 0)$.

    • The gradient of the line equals the constant of variation $k$.

    • If $x$ doubles, $y$ doubles; if $x$ is halved, $y$ is halved.

Pythagoras and Trigonometry

• Reviewing Pythagoras' Theorem (Consolidating)

  • Pythagoras' Theorem applies exclusively to right-angled triangles (triangles containing one $90^\circ$ angle).

  • Structural elements of a right-angled triangle:

    • Hypotenuse ($c$): The longest side of the triangle, always located opposite the $90^\circ$ right angle.

    • Legs ($a$ and $b$): The two shorter sides that meet to form the $90^\circ$ right angle.

  • Statement of the Theorem:         In any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

  • Formula:         $a^2 + b^2 = c^2$

  • Solving for the hypotenuse ($c$):         $c = \sqrt{a^2 + b^2}$

  • Pythagorean Triples: Sets of three positive integers $(a, b, c)$ that satisfy $a^2 + b^2 = c^2$. Standard primitive triples include:

    • $3, 4, 5$ ($3^2 + 4^2 = 9 + 16 = 25 = 5^2$)

    • $5, 12, 13$ ($5^2 + 12^2 = 25 + 144 = 169 = 13^2$)

    • $8, 15, 17$ ($8^2 + 15^2 = 64 + 225 = 289 = 17^2$)

    • $7, 24, 25$ ($7^2 + 24^2 = 49 + 576 = 625 = 25^2$)

• Finding the Length of a Shorter Side (Consolidating)

  • To find the length of an unknown shorter side ($a$ or $b$) when the hypotenuse ($c$) and one other side are known, rearrange Pythagoras' theorem by subtracting the square of the known leg from the square of the hypotenuse:         $a^2 = c^2 - b^2 \implies a = \sqrt{c^2 - b^2}$         $b^2 = c^2 - a^2 \implies b = \sqrt{c^2 - a^2}$

  • Step-by-step calculation procedure:

    1. Identify and label the hypotenuse $c$ opposite the right angle.

    2. Assign labels $a$ and $b$ to the remaining two sides, where one is unknown.

    3. Substitute the known numerical values into the rearranged formula $a = \sqrt{c^2 - b^2}$.

    4. Square the side values, compute the difference, and evaluate the square root.

• Applications of Pythagoras' Theorem

  • Practical problem-solving steps using Pythagoras' theorem:

    1. Extract spatial information from a contextual situation and construct a clear, labeled 2D diagram.

    2. Locate or construct right-angled triangles within the spatial layout.

    3. Convert all physical measurement values to uniform units (e.g., converting $cm$ to $m$) prior to calculation.

    4. Apply $c = \sqrt{a^2 + b^2}$ or $a = \sqrt{c^2 - b^2}$ to compute the desired unknown length.

    5. Round the calculated answer to the requested level of accuracy (or retain exact surd/radical form if specified), including appropriate units.

• Trigonometric Ratios (Consolidating)

  • Trigonometry relates the internal angles of a right-angled triangle to the ratios of its side lengths.

  • Side labeling relative to a reference acute angle $\theta$:

    • Hypotenuse ($H$): The side opposite the right angle (always the longest side).

    • Opposite ($O$): The side directly across from the reference angle $\theta$.

    • Adjacent ($A$): The side next to the reference angle $\theta$ that is not the hypotenuse.

  • The primary trigonometric ratios (SOH CAH TOA):

    • Sine ratio:             $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H}$

    • Cosine ratio:             $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H}$

    • Tangent ratio:             $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A}$

• Finding Unknown Side Lengths

  • Procedure for determining an unknown side when given an acute angle $\theta$ and one side length:

    1. Label the sides of the right-angled triangle as $O$, $A$, and $H$ relative to the given angle $\theta$.

    2. Select the trigonometric ratio that involves the given side and the unknown side.

    3. Set up the equation substituting the known angle and side length.

    4. Solve for the unknown side variable when it appears in the numerator:

       • If $\sin(\theta) = \frac{x}{H}$, then $x = H \times \sin(\theta)$.

       • If $\cos(\theta) = \frac{x}{H}$, then $x = H \times \cos(\theta)$.

       • If $\tan(\theta) = \frac{x}{A}$, then $x = A \times \tan(\theta)$.

• Solving for the Denominator

  • When the unknown variable $x$ appears in the denominator of a trigonometric equation, algebraic isolation requires a two-step rearrangement:

    • For the sine ratio:             $\sin(\theta) = \frac{O}{x} \implies x \times \sin(\theta) = O \implies x = \frac{O}{\sin(\theta)}$

    • For the cosine ratio:             $\cos(\theta) = \frac{A}{x} \implies x \times \cos(\theta) = A \implies x = \frac{A}{\cos(\theta)}$

    • For the tangent ratio:             $\tan(\theta) = \frac{O}{x} \implies x \times \tan(\theta) = O \implies x = \frac{O}{\tan(\theta)}$

• Finding Unknown Angles

  • When two side lengths of a right-angled triangle are known, inverse trigonometric functions (arcsin, arccos, arctan) are used to compute the value of an unknown acute angle $\theta$:

    • Inverse sine function:             $\theta = \sin^{-1}\left(\frac{O}{H}\right)$

    • Inverse cosine function:             $\theta = \cos^{-1}\left(\frac{A}{H}\right)$

    • Inverse tangent function:             $\theta = \tan^{-1}\left(\frac{O}{A}\right)$

  • Angular units and conversion:

    • Angles are evaluated in degrees ($^\circ$) and can be further refined into minutes ($'$) and seconds ($''$).

    • Conversion factor: $1^\circ = 60'$ and $1' = 60''$.

• Angles of Elevation and Depression

  • Angle of Elevation:

    • The angle measured upward between a horizontal line of sight and the line of sight looking up to an object above the observer.

  • Angle of Depression:

    • The angle measured downward between a horizontal line of sight and the line of sight looking down to an object below the observer.

  • Geometric relationship:

    • Because horizontal sight lines are parallel, the angle of elevation from an observer at point A to an object at point B is equal to the angle of depression from point B to the observer at point A (alternate interior angles theorem).

• Direction and Bearings

  • Compass Bearings:

    • Expressed using a primary direction (North or South), an acute angle turn, and a secondary direction (East or West).

    • Examples: $\text{N}30^\circ\text{E}$ (start facing North, rotate $30^\circ$ toward East), $\text{S}45^\circ\text{W}$ (start facing South, rotate $45^\circ$ toward West).

  • True Bearings (Three-Figure Bearings):

    • Measured clockwise starting from True North ($000^\circ\text{T}$) through a full circle of $360^\circ$.

    • Always written using three digits preceding the degree symbol (e.g., $005^\circ\text{T}$, $045^\circ\text{T}$, $225^\circ\text{T}$).

  • Back Bearings:

    • The reciprocal bearing pointing in the exact opposite direction ($180^\circ$ shift).

    • Calculation rule for True Bearings:

      • If the original bearing $\theta < 180^\circ$: $\text{Back Bearing} = \theta + 180^\circ$

      • If the original bearing $\theta \ge 180^\circ$: $\text{Back Bearing} = \theta - 180^\circ$

• Applications in Three Dimensions

  • Solving spatial problems in three-dimensional geometric structures (cubes, rectangular prisms, pyramids, cones):

    1. Identify key 3D diagonal segments and right angles within the structure.

    2. Break the 3D problem into a series of connected 2D right-angled triangular cross-sections.

    3. Apply 2D Pythagoras' theorem to find an intermediate side length on a base or face plane.

    4. Use the intermediate length in a second right-angled triangle containing the desired 3D diagonal or inclination angle.

  • Three-Dimensional Pythagorean Formula (for a rectangular box with length $x$, width $y$, and height $z$):         $d = \sqrt{x^2 + y^2 + z^2}$         where $d$ represents the space diagonal passing through the interior center of the box.

• The Sine Rule

  • The Sine Rule applies to any arbitrary non-right-angled (oblique) triangle with vertices $A, B, C$ and corresponding opposite side lengths a, b, c$.\n * Formula for calculating unknown side lengths:\n        \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\n * Formula for calculating unknown angles:\n        \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}\n * Conditions for applying the Sine Rule:\n * AAS / ASA: Given two angles and any one side length.\n * SSA (Ambiguous Case): Given two side lengths and a non-included angle opposite one of those sides.\n * The Ambiguous Case (SSA):\n * When given two sides a, b$and an acute angle$A$opposite side$a:\n * If a < b\sin(A): No valid triangle exists.\n * If a = b\sin(A): One right-angled triangle exists.\n * If a \ge b: One unique triangle exists.\n * If b\sin(A) < a < b$: Two distinct valid triangles exist—one acute triangle and one obtuse triangle where the obtuse angle is$B_2 = 180^\circ - B_1$.

• The Cosine Rule

  • The Cosine Rule relates all three sides of an oblique triangle to the cosine of one of its angles.

  • Formula for calculating an unknown side length (given two sides and the included angle, SAS):         $a^2 = b^2 + c^2 - 2bc\cos(A)$         $b^2 = a^2 + c^2 - 2ac\cos(B)$         $c^2 = a^2 + b^2 - 2ab\cos(C)$

  • Rearranged formula for calculating an unknown angle (given all three sides, SSS):         $\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}$         $\cos(B) = \frac{a^2 + c^2 - b^2}{2ac}$         $\cos(C) = \frac{a^2 + b^2 - c^2}{2ab}$

• Area of a Triangle

  • Standard formula (requires perpendicular height $h$):         $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2}bh$

  • Trigonometric Area Formula (does not require perpendicular height; requires two sides and the included angle, SAS):         $\text{Area} = \frac{1}{2}ab\sin(C)$         $\text{Area} = \frac{1}{2}bc\sin(A)$         $\text{Area} = \frac{1}{2}ac\sin(B)$