One Circle, Every Angle Trigonometric Functions

Trigonometry began, as the word itself says, as 'triangle measurement', a tool sea captains, surveyors, and engineers used to find distances and heights from angles inside a right-angled triangle. But a right-angled triangle can only ever contain an angle strictly between 0 and 90 degrees; ask for the sine of 200 degrees, or of -50 degrees, or of an angle so large it has wound all the way around a full circle and kept going, and a triangle has nothing to offer. This chapter rebuilds sine, cosine, and their relatives from a completely different starting point, a unit circle, one with radius exactly 1, so that they become genuine functions of any real number at all, not just a measurement trapped inside one shape.

Degrees split a full revolution into 360 equal parts, an arbitrary but familiar choice. Radians measure an angle by a completely different, more natural yardstick: the angle subtended at the centre of a circle by an arc exactly as long as the radius is called one radian. Since the circumference of a unit circle is exactly 2 pi (the radius being 1), one complete revolution measures exactly 2 pi radians, giving the fundamental conversion pi radians = 180 degrees. More generally, in a circle of radius r, an arc of length l subtends an angle theta (in radians) given by theta = l/r, or equivalently l = r theta, the single formula the rest of this chapter and the two that follow it in this thread quietly depend on. Convert 40 degrees 20 minutes to radians: 40°20' = 40 and 1/3 degrees, times pi/180, giving 121 pi/540 radians. Convert 6 radians to degrees: 6 times 180/pi, using pi is approximately 22/7, gives approximately 343 degrees 38 minutes 11 seconds. Radians look less friendly at first than degrees, but they are the measure every later formula in this thread, derivatives and integrals of trigonometric functions included, is built to expect.

Place a circle of radius 1 at the origin. For any real number x, positive or negative, mark off an arc of length x around the circle starting from the point (1, 0), travelling anticlockwise for positive x and clockwise for negative x, and let (a, b) be the point this arc ends at. Define cos x = a and sin x = b. Because (a, b) sits on a circle of radius 1, it must satisfy a squared + b squared = 1, giving the single most useful identity in the whole subject: cos squared x + sin squared x = 1, true for every real x, not just acute angles inside a triangle. The four other trigonometric functions are then defined directly in terms of these two: tan x = sin x / cos x (undefined wherever cos x = 0), cot x = cos x / sin x, sec x = 1/cos x, and cosec x = 1/sin x. Because winding once all the way around the unit circle (an angle of 2 pi) brings you back to exactly the same point, sin and cos automatically repeat every 2 pi: sin(2n pi + x) = sin x and cos(2n pi + x) = cos x, for any integer n. This single circular construction is what finally lets 'the sine of 750 degrees' or 'the cosine of -10' mean something concrete.

Moving around the unit circle passes through four quadrants, and the coordinates (a, b) change sign in each: both positive in the first quadrant, a negative and b positive in the second, both negative in the third, a positive and b negative in the fourth. Since cos x = a and sin x = b, this directly gives the sign of every trigonometric function in each quadrant, all of them derivable rather than memorised from scratch, though a short mnemonic (below) makes recall faster. Because -1 is less than or equal to a and b is less than or equal to 1 for every point on the unit circle, sin x and cos x are confined to the interval [-1, 1] for every real x, their domain is all real numbers and their range is exactly that interval. tan x and cot x, being ratios, have no such ceiling; their range is every real number, though each has specific angles (odd multiples of pi/2 for tan, integer multiples of pi for cot) where the denominator vanishes and the function itself is undefined. sec x and cosec x, as reciprocals of cos x and sin x, can never take any value strictly between -1 and 1, since dividing 1 by a number no bigger than 1 in size always gives a result at least 1 in size.

The formula for cos(x+y) looks like it should be cos x + cos y, but it is not, and the real formula, cos(x+y) = cos x cos y - sin x sin y, has a genuinely elegant proof using nothing but the unit-circle definition and the distance formula. Place four points on the unit circle: P1 at angle x, P2 at angle (x+y), P3 at angle -y, and P4 at angle 0 (the point (1,0)). The angle from P4 to P1 is x, and the angle from P3 to P2 is also x (since P2 is at x+y and P3 is at -y, the angle between them is (x+y)-(-y) = x+2y... actually the key insight is triangles P1OP3 and P2OP4 are congruent, since both have two sides of length 1 (radii) and the angle between them is x in both cases). Congruent triangles have equal corresponding sides, so the straight-line distance P1P3 equals the straight-line distance P2P4. Writing out both distances using the coordinate distance formula and the coordinates each point has by definition (cos and sin of its own angle), then setting the two expressions equal and simplifying, the cos x cos y and sin x sin y terms fall out directly, giving cos(x+y) = cos x cos y - sin x sin y with no trigonometric assumption beyond the unit-circle definition itself. Every other sum/difference identity, cos(x-y), sin(x+y), sin(x-y), and the tan and cot versions, follows from this one formula by substituting -y for y or using the co-function relationships cos(pi/2 - x) = sin x and sin(pi/2 - x) = cos x.

Once cos(x+y) and sin(x+y) are established, setting y = x gives the double-angle formulas: cos 2x = cos squared x - sin squared x (which rewrites, using the Pythagorean identity, into two other equally useful forms, 2 cos squared x - 1 and 1 - 2 sin squared x), and sin 2x = 2 sin x cos x. Setting y = 2x and substituting the double-angle results gives the triple-angle formulas: sin 3x = 3 sin x - 4 sin cubed x, and cos 3x = 4 cos cubed x - 3 cos x. A separate, equally useful family turns a SUM of two sines or cosines into a PRODUCT (and vice versa), found by adding and subtracting the cos(x+y) and cos(x-y) expansions: cos x + cos y = 2 cos[(x+y)/2] cos[(x-y)/2], and similarly for cos x - cos y, sin x + sin y, and sin x - sin y. These sum-to-product identities are exactly the tool that turns an intimidating expression like sin 5x + sin 3x into a compact product, 2 sin 4x cos x, often the single move that makes an otherwise unmanageable identity provable in a few lines.

The book's own history here is worth taking at face value rather than treating as a footnote: trigonometry as a systematic subject was first developed in India, not Greece. Aryabhata (476 CE) is credited with the earliest table of sine values; Brahmagupta (598 CE), Bhaskara I (600 CE), and Bhaskara II (1114 CE) extended and refined the results across several centuries, including Bhaskara II's exact expressions for the sine and cosine of 18, 36, 54, and 72 degrees. This body of knowledge travelled from India to the Arab world and from there into Europe; the book notes plainly that the Greeks' own competing approach to the subject was, by comparison, clumsy enough that it was set aside once the Indian methods became known. The word 'sine' itself descends from the Sanskrit astronomical term used in these siddhanta texts. Separately, the Greek philosopher Thales (around 600 BCE) is remembered for a different, equally clever trigonometric trick unrelated to sine tables at all: measuring the height of a great pyramid in Egypt purely from the length of its shadow and the shadow of a known-height staff planted beside it, using the fact that both shadows and heights share the same ratio, tan(the sun's angle in the sky). This is the very method Class 10's own Applications chapter, and this thread's third chapter still to come, both rely on.