Trigonometry

Introduction

Trigonometry literally means "triangle measurement." It grew out of practical problems — ancient astronomers needed to compute chord lengths, surveyors needed to find inaccessible distances, and navigators needed to chart courses across open ocean. Yet the subject rests on a beautifully simple idea: the ratios between the sides of a right-angled triangle depend only on its angles, not on its size. That single observation, which follows directly from the theory of similarity, connects trigonometry to almost every branch of mathematics and science.

This chapter develops trigonometry geometrically. We begin with the three basic ratios in a right triangle, derive exact values from constructions you already know, then extend the definitions to any angle using the unit circle. From there we prove the sine rule, the cosine rule, and the area formula for a general triangle — tools that free us from the right-angle restriction entirely.

Right-Triangle Trigonometry

Consider a right-angled triangle with one of its acute angles labelled theta. The side opposite the right angle is the hypotenuse — always the longest side. Relative to theta, one of the remaining sides lies opposite the angle and the other lies adjacent to it.

The three primary trigonometric ratios are defined as follows:

sin theta = opposite / hypotenuse

cos theta = adjacent / hypotenuse

tan theta = opposite / adjacent

The mnemonic SOH CAH TOA captures these three definitions: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.

Why do the ratios depend only on the angle?

This is a direct consequence of similarity. Any two right triangles sharing the same acute angle theta are similar (by AA similarity — they share the right angle and the angle theta). In similar triangles, corresponding sides are proportional. Therefore the ratio of opposite to hypotenuse is the same in every right triangle with angle theta, regardless of the triangle's size. The same reasoning applies to the cosine and tangent ratios. Trigonometry is, at its heart, a catalogue of similarity ratios.

Reciprocal ratios

Three further ratios are sometimes used:

cosec theta = 1 / sin theta = hypotenuse / opposite

sec theta = 1 / cos theta = hypotenuse / adjacent

cot theta = 1 / tan theta = adjacent / opposite

These reciprocal functions appear frequently in calculus and physics, though the three primary ratios are sufficient for all the work in this chapter.

Exact Values for 30, 45, and 60 Degrees

Rather than relying on a calculator, the trigonometric values at three special angles can be derived from simple geometric constructions.

The 45-degree angle

Take a square with side length 1 and cut it along a diagonal. The result is an isosceles right triangle — two sides of length 1 and a hypotenuse of length √2 (by Pythagoras' theorem). The two acute angles are each 45 degrees. Reading off the ratios:

sin 45 = 1 / √2 = √2 / 2

cos 45 = 1 / √2 = √2 / 2

tan 45 = 1 / 1 = 1

Notice that sine and cosine are equal at 45 degrees. This makes geometric sense: the triangle is isosceles, so opposite and adjacent are interchangeable.

The 30-degree and 60-degree angles

Begin with an equilateral triangle of side length 2. All angles are 60 degrees. Drop a perpendicular from one vertex to the opposite side; by symmetry it bisects both the base and the apex angle. This produces two congruent right triangles, each with angles 30, 60, and 90 degrees, a hypotenuse of 2, a short side of 1, and a long side of √3 (again by Pythagoras).

From the 30-degree angle:

sin 30 = 1 / 2

cos 30 = √3 / 2

tan 30 = 1 / √3 = √3 / 3

From the 60-degree angle:

sin 60 = √3 / 2

cos 60 = 1 / 2

tan 60 = √3

A useful pattern emerges: sin 30 = cos 60 and sin 60 = cos 30. In general, sin theta = cos(90 - theta). The sine of any angle equals the cosine of its complement — which is, in fact, the origin of the word "cosine" (complement's sine).

Summary table

These values recur throughout geometry, physics, and engineering. Committing them to memory — or better, knowing how to reconstruct them from the equilateral triangle and the square — is invaluable.

The Unit Circle

Right-triangle definitions work beautifully for acute angles (between 0 and 90 degrees), but many applications require angles of any size — 120 degrees, 200 degrees, negative angles, or angles measured in radians. The unit circle provides the generalisation.

Construction

Draw a circle of radius 1 centred at the origin of a coordinate plane. Take a ray from the origin making an angle theta with the positive x-axis, measured anticlockwise. The ray intersects the circle at a point P. The coordinates of P define the trigonometric functions:

cos theta = x-coordinate of P

sin theta = y-coordinate of P

tan theta = sin theta / cos theta = y / x (when x is not zero)

When theta is acute, this agrees with the right-triangle definitions: drop a perpendicular from P to the x-axis and you have a right triangle with hypotenuse 1 (the radius), adjacent side equal to the x-coordinate, and opposite side equal to the y-coordinate. But the unit-circle definition works for every angle — the coordinates of P are always well-defined.

The four quadrants

As the angle theta increases from 0 to 360 degrees, the point P traces out the full circle, and the signs of sine and cosine change depending on the quadrant:

• First quadrant (0 to 90 degrees): x > 0 and y > 0, so both sin and cos are positive.
• Second quadrant (90 to 180 degrees): x < 0 and y > 0, so sin is positive, cos is negative.
• Third quadrant (180 to 270 degrees): x < 0 and y < 0, so both sin and cos are negative.
• Fourth quadrant (270 to 360 degrees): x > 0 and y < 0, so cos is positive, sin is negative.

Since tan = sin / cos, the tangent is positive in the first and third quadrants and negative in the second and fourth. The mnemonic ASTC ("All, Sine, Tangent, Cosine") records which functions are positive in each quadrant.

Radians

Angles can also be measured in radians, where one full revolution equals 2 pi radians. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Key conversions: 180 degrees = pi radians, so 30 degrees = pi/6, 45 degrees = pi/4, 60 degrees = pi/3, and 90 degrees = pi/2. Radians are the natural unit for the unit circle and are standard throughout higher mathematics and physics.

The Sine Rule

The sine rule relates the sides of any triangle to the sines of the opposite angles. If a triangle has sides a, b, c opposite angles A, B, C respectively:

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

Proof

Consider a triangle ABC with an altitude h drawn from vertex B to side AC. In the right triangle formed on the left, h = c sin A. In the right triangle formed on the right, h = a sin C. Setting these equal:

c sin A = a sin C

Dividing both sides by sin A sin C gives a / sin A = c / sin C. Drawing an altitude from a different vertex yields the analogous result for side b, completing the proof.

The sine rule is particularly useful when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (the ambiguous case, SSA).

The ambiguous case

When using the sine rule with two sides and a non-included angle, there may be zero, one, or two valid triangles. This occurs because the sine function gives the same value for supplementary angles: sin theta = sin(180 - theta). Careful geometric reasoning — or a quick sketch — resolves the ambiguity.

The Cosine Rule

The cosine rule generalises the Pythagorean theorem to triangles that are not right-angled:

c squared = a squared + b squared - 2ab cos C

When C = 90 degrees, cos C = 0 and the formula reduces to c squared = a squared + b squared — Pythagoras' theorem. The cosine rule is therefore a strict generalisation: Pythagoras is a special case.

Proof

Place the triangle in a coordinate system with vertex C at the origin and side a along the positive x-axis. Then vertex A is at the point (b cos C, b sin C) and vertex B is at (a, 0). Applying the distance formula:

c squared = (a - b cos C) squared + (b sin C) squared

Expanding:

c squared = a squared - 2ab cos C + b squared cos squared C + b squared sin squared C

Since cos squared C + sin squared C = 1, the last two terms simplify to b squared:

c squared = a squared + b squared - 2ab cos C

This elegant proof shows how naturally trigonometry and coordinate geometry work together.

When to use the cosine rule

The cosine rule is the tool of choice when you know two sides and the included angle (SAS), or all three sides (SSS) and need to find an angle. In the latter case, rearrange:

cos C = (a squared + b squared - c squared) / 2ab

Area of a Triangle

The standard area formula — half base times height — requires the height, which is not always known. Trigonometry provides an alternative. If two sides a and b are known, together with the included angle C:

Area = (1/2) ab sin C

This follows immediately from the fact that the height relative to side a is b sin C (or vice versa). The formula is valid for both acute and obtuse triangles, since sin C is positive for all angles between 0 and 180 degrees.

This result is especially powerful in combination with the sine rule. If only one side and two angles are known, the sine rule can supply the missing sides, and the area formula completes the calculation.

Key Identities

The Pythagorean identity

The most fundamental trigonometric identity is:

sin squared theta + cos squared theta = 1

On the unit circle, sin theta and cos theta are the y- and x-coordinates of a point on a circle of radius 1. Pythagoras' theorem applied to that point gives x squared + y squared = 1, which is precisely the identity above. It holds for every angle, not just acute ones.

Dividing through by cos squared theta (when it is non-zero) gives:

tan squared theta + 1 = sec squared theta

Dividing instead by sin squared theta gives:

1 + cot squared theta = cosec squared theta

These three Pythagorean identities are indispensable in simplifying expressions and solving equations.

The tangent ratio

From the definitions:

tan theta = sin theta / cos theta

This is not a separate assumption — it follows directly from dividing the sine ratio by the cosine ratio, since the hypotenuse cancels. On the unit circle, tan theta is the slope of the radius to the point (cos theta, sin theta).

Symmetry and periodicity

The unit circle reveals several symmetry properties:

• Periodicity: sin(theta + 360) = sin theta and cos(theta + 360) = cos theta. The functions repeat every full revolution.
• Even and odd: Cosine is an even function: cos(-theta) = cos theta. Sine is an odd function: sin(-theta) = -sin theta.
• Complementary angles: sin(90 - theta) = cos theta and cos(90 - theta) = sin theta.
• Supplementary angles: sin(180 - theta) = sin theta and cos(180 - theta) = -cos theta.

These properties simplify calculations considerably and explain the patterns in the table of exact values.

Applications

Trigonometry is one of the most widely applied branches of mathematics. A few representative examples illustrate its reach.

Surveying and navigation

Surveyors measure angles with a theodolite and use the sine and cosine rules to calculate distances that cannot be measured directly — across rivers, between mountain peaks, or to inaccessible points. Triangulation, the method of fixing a position by measuring angles from known baseline points, was used historically to map entire countries and remains the foundation of modern geodesy.

Navigators use trigonometry to convert between bearings (angles measured clockwise from north) and Cartesian coordinates. The spherical counterparts of the sine and cosine rules — spherical trigonometry — are essential for calculating great-circle distances on the Earth's surface.

Physics and engineering

In physics, trigonometry appears whenever a quantity has direction. A force acting at an angle theta to the horizontal can be resolved into horizontal and vertical components using cosine and sine respectively:

F horizontal = F cos theta

F vertical = F sin theta

This decomposition of vectors is fundamental in mechanics, electromagnetism, and fluid dynamics. Simple harmonic motion — the oscillation of a pendulum or a vibrating spring — is described by sinusoidal functions. Sound waves, light waves, and alternating electrical currents are all modelled using sine and cosine, making trigonometry the mathematical language of wave phenomena.

Architecture and design

Architects use trigonometry to calculate roof pitches, ramp gradients, and the geometry of arches and domes. In computer graphics, rotations and projections rely heavily on sine and cosine. The transformation matrices that rotate objects on screen are built entirely from trigonometric functions.

Astronomy

Trigonometry began with astronomy, and it remains central to the discipline. The parallax method for measuring stellar distances, the calculation of orbital elements, and the prediction of eclipses all depend on trigonometric relationships. Hipparchus, often called the father of trigonometry, compiled the first known table of chords (the ancient equivalent of a sine table) precisely for astronomical calculations in the second century BCE.

Connection to Earlier Chapters

Trigonometry sits at a crossroads in geometry. It depends on ideas from several earlier chapters and provides tools that simplify and extend them.

From the chapter on triangles, we inherit the angle sum property (A + B + C = 180 degrees), the classification of triangles by their angles, and Pythagoras' theorem — which we have now generalised via the cosine rule. The congruence conditions (SAS, ASA, AAS, SSS) tell us when a triangle is uniquely determined; trigonometry tells us how to compute it.

From the chapter on similarity, we gain the justification for the trigonometric ratios themselves. It is precisely because similar triangles have proportional sides that sine, cosine, and tangent can be defined as functions of an angle alone.

From the chapter on the circle, we take the unit circle, which extends the trigonometric definitions to all angles. The inscribed angle theorem and the properties of arcs and chords find natural expression in the language of sine and cosine. Indeed, the word "sine" derives from a mistranslation of the Sanskrit word "jya" (bowstring — referring to a half-chord of a circle), through Arabic and Latin intermediaries.

Trigonometry in turn opens the door to further topics: coordinate geometry uses sine and cosine to define rotations and polar coordinates; the study of periodic phenomena in physics relies on trigonometric functions; and the geometry of three-dimensional space — including the platonic solids and the sphere — is greatly enriched by trigonometric methods.

Conclusion

Trigonometry transforms the qualitative geometry of triangles into a quantitative tool. With only an angle and a side, the sine and cosine rules can reconstruct an entire triangle. With the unit circle, the definitions extend seamlessly to every angle, laying the groundwork for wave theory, vector analysis, and higher geometry. The subject is a perfect illustration of how a simple geometric observation — that similar right triangles share the same ratios — can generate an entire branch of mathematics with applications across science and engineering.

The next chapter examines the precise geometric procedures for constructing figures with compass and straightedge — Euclidean Constructions.