Chapter 19 Algebra Logarithms Coordinate Geometry and Geological Trigonometry

Year 1 · Mathematics & Statistics · G117-L1

Apply equations, logarithms, coordinate geometry and trigonometry to a worked bed-thickness problem with clearly stated geometric assumptions.

A repeatable workflow from a testable question to an evidence-based interpretation. Original Geology Hub teaching diagram.

Learning objectives: Manipulate equations, interpret logarithms and graphs, resolve a geometric measurement, and check whether a mathematical solution fits geological conditions.

Algebra expresses a relationship before numbers enter

Variables represent quantities, and equations state relationships among them. Rearranging an equation preserves equality when the same valid operation is applied to both sides. For density ρ = m/V, mass is m = ρV. Units provide an independent check: density multiplied by volume should produce mass. A numerical answer without a consistent quantity and unit is incomplete.

Indices compactly describe repeated multiplication. A negative exponent indicates a reciprocal, and a fractional exponent can represent a root under suitable conditions. Logarithms answer an inverse question: if 10ˣ = y, then x = log₁₀y. Multiplication becomes addition of logarithms, and division becomes subtraction. The argument must be positive for a real logarithm. Logarithmic graph axes represent ratios, so equal spacing does not mean equal arithmetic increments.

Solve equations and test their physical meaning

A linear equation has a variable to the first power. A quadratic equation includes a squared term and can have two, one or no real roots. Simultaneous equations represent constraints that must hold together. Their solutions correspond geometrically to intersections: two lines, a line and a curve, or two curves. A mathematically valid root can be physically impossible, such as a negative duration in a problem defined only after an event begins.

Coordinate geometry links equations to shape. The distance between two planar points follows the Pythagorean relationship. A straight line has a slope and intercept in a chosen coordinate system. Circles and parabolas have characteristic equations, but fitting a curve to data does not establish its geological cause. Changing coordinates can simplify a problem, provided the transformation and reference axes remain documented.

Trigonometry turns angles into components

For a right triangle, sine relates opposite side to hypotenuse, cosine adjacent side to hypotenuse and tangent opposite to adjacent. These functions extend through the unit circle to general angles. Radians measure angle through arc length relative to radius; degrees divide a full turn into 360 parts. A calculator using the wrong angle mode can return a plausible-looking but incorrect result.

Trigonometric identities allow equivalent expressions, while sum, difference, multiple-angle and half-angle formulas simplify more complicated geometry. Inverse trigonometric functions recover an angle within specified ranges, so quadrant and orientation still require thought. A slope angle alone does not establish a compass direction. Geological geometry needs both magnitude and spatial orientation.

Powers and logarithms preserve relationships

The rule a^maⁿ = a^m+n explains why multiplying quantities on a logarithmic scale corresponds to adding their logarithms, within the relevant domain. A square root corresponds to a one-half power. For positive x, log(x²) = 2 log(x), while log(x + y) cannot generally be replaced by log(x) + log(y). This distinction matters when transforming measured geological variables. A straight trend after logarithmic transformation represents a different relationship from a straight trend on ordinary arithmetic axes.

Equation systems express simultaneous constraints

Two measurements can impose two equations on unknown quantities. Suppose fictional component masses satisfy x + y = 10 grams and 2x + y = 14 grams. Subtraction gives x = 4 grams, then y = 6 grams. A linear relation combined with a quadratic can yield more than one intersection. Two quadratics can also have several candidate solutions. Substitute each result into both original equations and reject values outside the physical domain rather than treating the first algebraic root as automatically correct.

Trigonometric identities connect alternative descriptions

The identity sin²θ + cos²θ = 1 connects the components of a unit-radius direction. Sum and difference formulas describe compound angles; for example, sin(A + B) combines sin A cos B and cos A sin B. Double-angle and half-angle forms follow related geometric relationships. Inverse functions recover an angle only within a chosen range, so an inverse tangent of a component ratio does not by itself resolve the correct compass quadrant. The signs of both components remain necessary.

Coordinates turn shapes into equations

In a planar grid, a straight line can be represented as y = mx + b, with slope m and intercept b. A circle centred at the origin satisfies x² + y² = r². A parabola has a quadratic form in an appropriate coordinate setting. Translating the origin changes coordinate values without changing the physical objects; rotating axes changes the components used to describe a direction. These transformations simplify geometry only when the old and new references are explicitly related.

Apply geometry with an interpretable sketch

A hypothetical geological surface rises 10 metres over 20 horizontal metres along its steepest direction. Its inclination is arctan(10/20), approximately 26.6 degrees. The ratio 0.5 is a gradient, not an angle of 0.5 degrees. If the traverse is oblique to the steepest direction, the observed inclination is an apparent dip and needs further geometry to recover true dip. The sketch establishes which distances belong in the triangle; the trigonometric operation then translates that already defined relationship into a numerical angle.

Worked true-thickness calculation

Imagine two parallel bed contacts cropping out on a horizontal surface along a line perpendicular to strike. Their map separation is 80 metres, and the bed dips at 30 degrees. In this deliberately simple geometry, true thickness equals horizontal separation multiplied by sin dip: 80 × sin 30° = 40 metres. The perpendicular-to-strike and horizontal-surface conditions are essential. A line oblique to strike or an inclined ground surface requires a different construction.

Check limiting behaviour. For a very small dip, broad horizontal separation can correspond to a small perpendicular thickness. For a vertical bed, horizontal separation in the correct direction equals thickness. A diagram should make these cases understandable before a formula is memorized. Then rearrange the relation to calculate separation for a specified thickness.

As practice, solve x² − 5x + 6 = 0 by factoring. The roots are two and three. If x represented an intersection distance, determine from the diagram whether both roots are relevant. The completed worksheet should include equations, units, a labelled sketch and a written reason for accepting or rejecting each mathematical solution.

Self-check

What does log₁₀1000 equal?

It equals 3 because 10³ = 1000.

Why must angle mode be checked?

Degree and radian inputs represent different angles unless converted correctly.

What assumption allows thickness = map width × sin dip here?

Contacts are parallel, the surface is horizontal and width is measured perpendicular to strike.

Sources and further study

Topic framework: the supplied University of Peshawar BS Geology curriculum, session 2004–05 onward. This independently authored learning resource is not affiliated with the university and does not confer a degree or accreditation.

Chapter 19 multiple choice questions

Choose one best answer for each question. Open each answer panel after making your choice.

A value is plotted at log₁₀ y = 3. What is y?
1. 1,000
2. 3
3. 30
4. 100
Answer and explanation
A. 1,000 A logarithm of three means that 10 raised to the third power equals the original value.
Two parallel contacts are 80 m apart along a horizontal line perpendicular to strike. The bed dips 30°. What is the true thickness under this geometry?
1. 80 m
2. 160 m
3. 40 m
4. Approximately 69.3 m
Answer and explanation
C. 40 m True thickness is horizontal separation multiplied by sin dip: 80 × 0.5 = 40 m.
Why would the same thickness formula need reconsideration if the traverse were oblique to strike?
1. Obliquity changes the mineral composition of the bed.
2. The measured horizontal separation would have a different geometric relationship to true thickness.
3. The dip would necessarily become zero.
4. Every oblique traverse equals true thickness directly.
Answer and explanation
B. The measured horizontal separation would have a different geometric relationship to true thickness. The stated formula assumes a horizontal measurement perpendicular to strike.
A quadratic model gives roots of −2 years and 3 years for an event defined only after time zero. Which solution is physically admissible?
1. −2 years only
2. Both roots must be accepted regardless of the time definition.
3. Neither root can ever describe an event.
4. 3 years only
Answer and explanation
D. 3 years only Mathematical solutions must be checked against the defined physical domain.
A student enters 30 into a calculator set to radians when solving a problem with 30°. What is the main issue?
1. Sine and cosine become identical functions.
2. The geological bed changes its real orientation.
3. The calculator interprets a different angle unless the units are converted.
4. The numerical result becomes independent of angle.
Answer and explanation
C. The calculator interprets a different angle unless the units are converted. Degrees and radians are different angle units, so the input mode must match the problem.

All 84 chapters

Search This Blog

Geology Hub