Trigonometry Basics

Part of Mathematics

The branch of mathematics dealing with triangles and angles — essential for surveying, navigation, construction, and astronomy.

Why This Matters

Trigonometry is how you extract useful measurements from angles. It is the bridge between the world of angles (which you can measure with a simple protractor or shadow observation) and the world of distances (which you often cannot directly measure). A surveyor who cannot cross a river can still measure the river’s width by taking two angle measurements from this side. A navigator who knows their heading and speed can calculate their position. A builder who measures the angle of a hillside can calculate the load on retaining walls.

More broadly, trigonometry is what elevates a society from “measuring only what you can reach with a stick” to “measuring everything you can see.” This transition — from local to remote measurement — is one of the key conceptual advances that separates early agricultural societies from early civilizations. The Babylonians had trigonometry. The Greeks formalized it. Arab mathematicians expanded it. European explorers used it to map the world. Every one of these developments depended on the same core insight: the ratios between sides in a right triangle depend only on the angle, not on the triangle’s size.

This article covers the conceptual foundations of trigonometry, how to measure angles without instruments, the sine rule and cosine rule for non-right triangles, and key applications in surveying and construction.

Angles: Measurement and Tools

All trigonometry begins with measuring angles. An angle is the rotation between two lines meeting at a point. The standard unit is degrees (°): a full rotation = 360°, a straight line = 180°, a right angle = 90°.

Measuring Angles Without a Protractor

The shadow clock method: At any time of day, a vertical stick (gnomon) casts a shadow. The angle of the sun above the horizon can be calculated from the stick height and shadow length: angle = arctan(height/shadow length).

The plumb-bob angle finder: Tie a plumb bob to a straight board. Mark the point where the string crosses a curved scale at the board’s end. Calibrate by marking when the board is level (plumb line hits 0°), then at known tilts (45° = line bisects the right angle).

Making a protractor: On paper or bark, draw a semicircle with a ruler-drawn diameter. Divide the semicircle into 180 equal parts. For basic construction accuracy:

1. Draw diameter AB
2. Construct perpendicular at center C (bisect AB) — this gives 90° marks
3. Bisect each 90° quarter to get 45° marks
4. Construct 60° by compass (equilateral triangle method) — gives 60° and 120°
5. Bisect these to get 30°, 15°, 75°, etc.
6. For angles not at these landmarks, subdivide by eye to the nearest degree

A careful craftsperson can achieve 1–2 degree accuracy with this method.

Angle Measurement in Practice

Angle of Elevation and Depression

The angle of elevation is the angle between horizontal and the line of sight upward to an object. The angle of depression is the angle below horizontal. Measuring these:

1. Construct a simple clinometer: a protractor with a plumb bob hanging from the center
2. Sight along the straight edge toward the object
3. Read the angle where the plumb line crosses the scale (subtract from 90° for elevation above horizontal)

Alternatively, use a stick and a knot of string: hold the stick level (use a water level or plumb to verify), hold the end of the stick toward the object, and use a protractor to measure the angle the stick makes relative to the line of sight.

The Four Quadrants

Beyond the first quadrant (0° to 90°), angles continue:

• First quadrant (0–90°): All trig ratios positive
• Second quadrant (90–180°): Sin positive, cos and tan negative
• Third quadrant (180–270°): Tan positive, sin and cos negative
• Fourth quadrant (270–360°): Cos positive, sin and tan negative

For construction and surveying, you mostly work in the first quadrant. Navigation and astronomy require all four.

Bearing vs. angle: A bearing is measured clockwise from north (0–360°). An angle in standard math notation is measured counterclockwise from east. Converting: math angle = 90° − bearing (for bearings 0–90°), with adjustments for other quadrants.

The Sine Rule

For any triangle (not just right triangles), with angles A, B, C and opposite sides a, b, c:

a / sin(A) = b / sin(B) = c / sin(C)

Use when: You know one side and two angles, or two sides and a non-included angle.

Example: Surveying across a river

You are on the north bank of a river. You mark two points (P and Q) 100 m apart along the bank. From P, you measure the angle to a tree T on the south bank: angle TPQ = 70°. From Q, angle TQP = 65°.

• The third angle at T: 180° − 70° − 65° = 45°
• Known side: PQ = 100 m, opposite angle T = 45°
• Find PT: PT / sin(65°) = 100 / sin(45°)
  • PT = 100 × sin(65°) / sin(45°) = 100 × 0.906 / 0.707 = 128 m
• Find QT: QT / sin(70°) = 100 / sin(45°)
  • QT = 100 × sin(70°) / sin(45°) = 100 × 0.940 / 0.707 = 133 m

You now know the distance to the tree from both bank points. You can use this to triangulate the tree’s exact position and calculate the river width.

The Cosine Rule

For any triangle, with the same notation:

c² = a² + b² − 2ab × cos(C)

Use when: You know three sides (and want an angle) or two sides and the included angle (and want the third side).

Example: Finding an angle

A triangular field has sides 80 m, 60 m, and 50 m. What is the angle opposite the 80 m side?

• a = 80, b = 60, c = 50; find angle A (opposite side a)
• a² = b² + c² − 2bc × cos(A)
• 6400 = 3600 + 2500 − 2(60)(50) × cos(A)
• 6400 = 6100 − 6000 × cos(A)
• 300 = −6000 × cos(A)
• cos(A) = −300/6000 = −0.05
• From trig table: cos(90°) = 0, cos(95°) ≈ −0.087; interpolating: A ≈ 93°

Note: negative cosine means the angle is obtuse (greater than 90°). This is expected — the 80 m side is the longest.

Note: The Pythagorean theorem is a special case of the cosine rule when C = 90°: cos(90°) = 0, so c² = a² + b² (the standard form).

Triangulation: The Core Surveying Method

Triangulation is the systematic use of trigonometry to measure large distances and map territory.

Principle:

1. Establish a baseline of known length (measured directly)
2. From each end of the baseline, measure angles to distant points
3. Use the sine rule to calculate distances to those points
4. Use those points as new baselines for further triangulation

Example network:

• Baseline AB = 500 m (measured with rope and stakes)
• Point C (hilltop) visible from both A and B
• Angle at A = 55°, angle at B = 62°
• Third angle at C = 180° − 55° − 62° = 63°
• AC = 500 × sin(62°) / sin(63°) = 500 × 0.883 / 0.891 = 495 m
• BC = 500 × sin(55°) / sin(63°) = 500 × 0.819 / 0.891 = 460 m

Now you know the distance to the hilltop from both baseline points. Use the hilltop as a new control point for the next triangle. Repeat across the entire territory to build a map.

Height by Angle

The most common single-triangle trig application: finding the height of something you cannot climb.

Procedure:

1. Stand at a measured distance D from the base of the object
2. Measure the angle of elevation θ to the top
3. Height = D × tan(θ)

Example: D = 30 m, θ = 42°

• Height = 30 × tan(42°) = 30 × 0.900 = 27 m

For tall objects where you cannot get close enough: Measure from two distances D₁ and D₂, get angles θ₁ and θ₂, and solve:

• Height = (D₂ − D₁) × tan(θ₁) × tan(θ₂) / (tan(θ₁) − tan(θ₂))

This works even when the base of the object is not accessible.

Building a Basic Trig Reference

For field use, record on durable material:

• SOH-CAH-TOA (see Trig Ratios)
• Table of sin, cos, tan for 0°, 15°, 30°, 45°, 60°, 75°, 90°
• Sine rule: a/sin(A) = b/sin(B)
• Cosine rule: c² = a² + b² − 2ab·cos(C)
• Height formula: H = D × tan(θ)

With these five items and a way to measure angles, you can solve virtually every surveying, construction, and navigation problem that requires trigonometry.

Trigonometry is not advanced mathematics — it is the natural extension of the right-triangle geometry you already use for construction layout, generalized to cover all possible triangle shapes and measurement scenarios. A community with even one person competent in its use gains an enormous practical advantage in land survey, construction planning, and navigation.