Geometry, Euclid and Trigonometry

Geometry is where mathematics first learned to prove things, and trigonometry is where geometry learned to measure. This page follows the path from similar triangles through the unit circle to precalculus, and adds an optional track for reading Euclid and Apollonius by reconstructing their arguments rather than merely reading them.

Why geometry comes before trigonometry

Demonstrated Trigonometric ratios are a consequence of similarity: two right triangles with the same acute angle have proportional sides, so the ratio of opposite to hypotenuse depends only on the angle. That is the definition of sine. A learner who skips similarity and proportion tends to treat sine as a calculator button, and then cannot tell why the laws of sines and cosines hold.

The core early results are:

• Similarity and proportion. Equal angles give proportional sides (Elements VI.4).
• The Pythagorean theorem and its converse. In a right triangle a^2 + b^2 = c^2 (Elements I.47). The converse (I.48) says that if a triangle's sides satisfy this equation, the triangle is right-angled. Both directions matter: one derives a length, the other tests a shape.
• Coordinate distance and slope. Distance between two points is the Pythagorean theorem applied to the horizontal and vertical differences, which connects geometry to the algebra in Algebra: The Skills That Carry Everything.
• Circles, angles and area. Area needs the perpendicular height, not the slanted side, and answers are in square units. Radians define an angle as arc length divided by radius, so a full turn is 2 pi radians.

Trigonometry on the unit circle

On a circle of radius 1, the point at angle t has coordinates (cos t, sin t). That single picture gives the values and signs a learner should be able to reconstruct rather than memorise: sin 30 degrees = 1/2, cos 60 degrees = 1/2, sin 45 degrees = sqrt(2)/2, and the sign pattern by quadrant.

Graphs of y = A sin(B(x - C)) + D have amplitude |A|, period 2 pi/|B|, a phase shift C and a vertical shift D. These are the same parameters that describe any simple wave.

Fact versus analogy. It is a physical fact that an ideal vibrating string produces a sum of sinusoidal modes with frequencies in whole-number ratios (the harmonic series). It is a historical and pedagogical analogy, not a theorem, to say that Greek ratio-geometry "contains" this physics. You can explore the ratios on the monochord and the way they fail to close in the Pythagorean comma.

Laws, identities and equations

For a triangle with sides a, b, c opposite angles A, B, C:

• Law of sines: a / sin A = b / sin B = c / sin C.
• Law of cosines: c^2 = a^2 + b^2 - 2ab cos C. When C is a right angle it reduces to the Pythagorean theorem, which is a good check on your memory of it.

An identity such as sin^2 t + cos^2 t = 1 is true wherever both sides are defined; tan t = sin t / cos t carries the domain restriction cos t not equal to 0. A trigonometric equation such as sin t = 1/2 has infinitely many solutions in general (t = pi/6 + 2 pi k or 5 pi/6 + 2 pi k), so say whether you want all solutions or those in a stated interval.

Completing precalculus

Precalculus gathers what calculus will assume: composite and inverse functions (and why an inverse trigonometric function needs a restricted domain), conic sections as graphs of second-degree equations, vectors, matrices, sequences and series, and an informal first look at limits. It is the stage where a learner should be fluent enough in algebra and trigonometry that calculus difficulties are about calculus, not about arithmetic. The next step is Calculus and Beyond: Change, Structure and Uncertainty.

A track for Euclid and Apollonius

Established Euclid's Elements (about 300 BCE) is the classic demonstration that a large body of geometry follows from a few definitions, postulates and common notions. A hands-on track does not need to be a full course; a small regular block of study time is enough.

1. Book I. Begin with definitions, postulates and common notions (see Euclid's postulates), then propositions in dependency order. I.1 constructs an equilateral triangle on a given segment; the sequence builds to the parallel postulate, triangle area and the Pythagorean theorem at I.47.
2. Books II to IV. Circles and inscribed figures, plus Book II, traditionally called "geometric algebra" (a historians' label that is debated).
3. Books V and VI. Ratio and proportion, then similarity. Book V presents the theory of magnitudes traditionally attributed to Eudoxus, which handles incommensurable lengths without assuming every ratio is a fraction. Read with a commentary and do not quietly replace magnitudes by rational numbers.
4. Apollonius after modern conics. Once conic sections are familiar from coordinates, Apollonius's Conics shows the ancient synthetic treatment. T. L. Heath's 1896 edition is an edition in modern notation, now in the public domain, and it does not follow the ancient text proposition for proposition.

The study method is the same each time: draw and label the figure, list what the argument depends on, attempt the proof yourself, read the original, and reproduce it from memory days later. Compare a synthetic proof (from axioms and figures) with a coordinate derivation (from algebra). They establish the same result by different routes, and seeing both is instructive. For the logic underneath, see Proof and Precise Reasoning: From Arguments to Theorems; for the role of this study among the seven liberal arts, see The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium.

Ancient geometry enriches the main route; it should not hold up calculus indefinitely. One suggested target is six varied Euclidean arguments reconstructed and understood, which is a portfolio, not a completion of the whole work.

Why it matters beyond geometry

Euclid is also a model of how to know something: terms defined, assumptions stated, steps checked. That ties to a wider question explored in What Does It Take to Know Something?, and to the overview in A Map of Mathematics: Twenty-Five Areas and What Depends on What.

Try this

1. Prove for yourself that the base angles of an isosceles triangle are equal (Elements I.5), then write the dependency list of earlier propositions you used.
2. Draw a 3-4-5 triangle on graph paper and test the Pythagorean converse. Then check that 2-3-4 is not a right triangle using the law of cosines.
3. Without a calculator, sketch y = 2 sin(3x) and state its amplitude and period (2 and 2 pi/3).

Further reading

• Euclid, Elements, with D. E. Joyce's online edition and commentary (Clark University).
• T. L. Heath, The Thirteen Books of Euclid's Elements, 2nd ed. (Cambridge, 1925; Dover reprint).
• T. L. Heath, Apollonius of Perga: Treatise on Conic Sections (Cambridge, 1896).
• Stewart, Redlin and Watson, Precalculus: Mathematics for Calculus (Cengage).