Trigonometric Functions
The Unit Circle and the Six Trigonometric Functions
Defining sine and cosine as coordinates on the unit circle, the four functions derived from them, and why the definitions extend beyond acute angles.
What this page covers
- Define sine and cosine as coordinates on the unit circle
- Derive the other four functions from them
- Connect the circle definition to the right-triangle ratios
- State the domain and range of each function
The unit circle definition
Take the circle of radius 1 centred at the origin. Starting from the point (1, 0), travel anticlockwise around the circumference a distance t. The point you arrive at has coordinates that define the two basic functions.
Because the radius is 1, the coordinates are the functions — no division is needed. The source draws the point P(x, y) on the circle with cos t marked along the horizontal and sin t along the vertical.
The source's check that the radius is 1
For the point P(35, -45):
So P is genuinely on the unit circle, and sin t = -45, cos t = 35. The signs place it in the fourth quadrant.
This is worth fixing firmly. Cosine is the horizontal coordinate and sine the vertical one. Reversing them makes every subsequent identity come out wrong, and the error is easy to make because the alphabetical order suggests the opposite.
The other four
Everything else is built from x and y. The source gives the definitions in both forms — as coordinate ratios and as reciprocals.
| Function | In coordinates | In terms of sine and cosine |
|---|---|---|
| sin t | y | — |
| cos t | x | — |
| tan t | yx | sin tcos t |
| sec t | 1x | 1cos t |
| csc t | 1y | 1sin t |
| cot t | xy | 1tan t |
Week 7, page 2 gives sec t = 1y and csc t = 1x. These are swapped. Since x = cos t and sec t = 1cos t, secant must be 1x; and cosecant, being 1sin t, must be 1y. The right-hand column of the same page has it correct, so the error is confined to the coordinate form. The table above uses the corrected version.
A useful mnemonic for which reciprocal goes with which: the co in cosecant pairs it with sine, not cosine, and secant pairs with cosine. The names are deliberately crossed.
The right-triangle picture
For an acute angle the circle definition reduces to the familiar ratios. Drop a perpendicular from P to the horizontal axis and a right triangle appears, with legs x and y and hypotenuse r.
On the unit circle r = 1, so the ratios collapse to the coordinates themselves. That is the whole relationship: the triangle definition is the circle definition with the hypotenuse scaled to 1.
Sine = Opposite over Hypotenuse; Cosine = Adjacent over Hypotenuse; Tangent = Opposite over Adjacent. A serviceable mnemonic, but it applies only to acute angles in a right triangle. The circle definition is the general one.
Domain and range
Because x and y are coordinates on a circle of radius 1, neither can exceed 1 in size. That bounds sine and cosine immediately.
| Function | Domain | Range |
|---|---|---|
| sin t | All real t | [-1, 1] |
| cos t | All real t | [-1, 1] |
| tan t | t ≠ π2 + npi | All real values |
| sec t | t ≠ π2 + npi | |y| ≥ 1 |
| csc t | t ≠ npi | |y| ≥ 1 |
| cot t | t ≠ npi | All real values |
The source records the sine and cosine cases directly: domain (-∞, ∞) for both, range [-1, 1]. It writes the ranges with round brackets on page 3, but the endpoints are attained — sinπ2 = 1 exactly — so square brackets are correct.
The exclusions all come from a zero denominator. Tangent and secant divide by x = cos t, which vanishes at π2 and 3pi2; cosecant and cotangent divide by y = sin t, which vanishes at 0 and π.
| t | -π | -π2 | 0 | π2 | π | 3pi2 | 2pi |
|---|---|---|---|---|---|---|---|
| sin t | 0 | −1 | 0 | 1 | 0 | −1 | 0 |
| cos t | −1 | 0 | 1 | 0 | −1 | 0 | 1 |
| tan t | 0 | undefined | 0 | undefined | 0 | undefined | 0 |
Periodicity
Travelling a further 2pi around the circle returns to the same point, so every trigonometric function repeats.
Tangent repeats twice as often. Since tan t = yx and a half-turn negates both coordinates, the ratio is unchanged: tan(t + π) = tan t, giving period π.
Periodicity is why any trigonometric equation with a solution has infinitely many. Restricting to a stated interval such as 0 ≤ t ≤ 2pi is what makes the answer finite, and the source always states such an interval.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| x = sin t, y = cos t | The other way round | At t = 0 the point is (1, 0), so cos 0 = 1 |
| sec t = 1sin t | 1cos t | The co in cosecant pairs it with sine |
| sin t > 1 for some t | Never | It is a coordinate on a unit circle |
| tanπ2 given a value | Undefined | cosπ2 = 0 |
| Using SOHCAHTOA for an obtuse angle | Use the circle definition | There is no such right triangle |
| Giving one solution to a trigonometric equation | There are infinitely many | Periodicity |
Frequently asked questions
Why define them on a circle rather than a triangle?
Because a triangle only gives angles between 0 and 90°. The circle definition works for any angle, including negative ones and those beyond a full turn, and it agrees with the triangle definition where both apply.
Which is x and which is y?
x = cos t and y = sin t. Cosine is the horizontal coordinate. Getting these the wrong way round is the commonest error in the topic.
Why are tangent and secant undefined at some angles?
Both have x = cos t in the denominator, so they fail wherever cos t = 0 — at π2, 3pi2 and every angle differing from these by π.
How does the triangle definition fit in?
For an acute angle in a triangle of hypotenuse r, the coordinates scale: sinθ = yr and cosθ = xr. On the unit circle r = 1, so the coordinates are the ratios.
Source. Handwritten teaching notes, Week 7 page 2 and Week 8 pages 1, 3-4.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.