mathematics / trigonometry / how they are all related
How They Are All Related
Six functions, three identities and one table of exact values, all derived from a single right triangle with a hypotenuse of 1.
Step 1 of 6
One triangle
Every relationship in this module comes from one triangle with a hypotenuse of 1.
Take the triangle under the radius on the unit circle. The hypotenuse is 1, the vertical side is sin θ, and the horizontal side is cos θ. That is the whole apparatus.
Everything that follows on this page is that triangle with something done to it. Pythagoras applied to it gives the first identity. Scaling it so the horizontal side becomes 1 gives the second. Scaling it so the vertical side becomes 1 gives the third. Two particular shapes of it, taken from a square and from an equilateral triangle, give the exact values everyone is asked to memorise.
If you remember nothing else from this module, remember the picture rather than the formulas. Each formula can be re-derived from it in a few seconds, and a formula you can re-derive is one you cannot misremember.
How to read this
The unit circle with that triangle drawn under the radius, its vertical side labelled sin and its horizontal side labelled cos. Drag the point, or use the angle control at the top of the page, and every figure below moves with it. Watch the two shorter sides change while the hypotenuse holds at 1. The six values underneath are the same numbers, read off as text.
Black: the radius, always exactly 1. Amber solid: cos = 0.500. Teal solid: sin = 0.866.
1 unit = 136 px.
- sin 0.866 √3/2
- csc 1.155 2/√3
- cos 0.500 1/2
- sec 2.000 2
- tan 1.732 √3
- cot 0.577 1/√3
A solid swatch is the function, a dashed one its reciprocal. Teal is the sine family, amber the cosine family, purple the tangent family.
Step 2 of 6
Pythagoras on that triangle
Pythagoras on that triangle gives sin²θ + cos²θ = 1.
The triangle has legs sin θ and cos θ and a hypotenuse of 1. Pythagoras says the squares on the legs add up to the square on the hypotenuse:
sin²θ + cos²θ = 1
The notation sin²θ means (sin θ)², the sine squared. The exponent is written after the function name by convention, to keep it distinct from sin(θ²) and from the inverse function.
This is an , which means it is true for every value of θ, not an equation to be solved for a particular one. Test it anywhere. At 30 degrees, sin is 1/2 and cos is √3/2, so the squares are 1/4 and 3/4, which add to exactly 1. At 45 degrees both are √2/2, so both squares are 1/2, adding to exactly 1. At 0, sin is 0 and cos is 1, so 0 plus 1 is 1.
It also says something geometric. The point (cos θ, sin θ) is always exactly 1 away from the origin. That is what being on a circle of radius 1 means, and this identity is that statement written in algebra.
How to read this
One bar of length 1, split into the two squared areas sin²θ and cos²θ, with the arithmetic printed beside it. As θ moves, area flows from one part into the other and the total holds at 1. This is the first of the three bars in the next figure, shown on its own first because the other two are built from it. Beneath it the same fact is built up as a graph, in three steps. First the two ordinary waves, sin θ and cos θ, both swinging between −1 and 1. Then both are squared, which folds everything below the axis upwards and leaves both curves between 0 and 1. Then the cosine curve is hung from the top of the frame instead of standing on the floor. Because cos²θ is 1 minus sin²θ, the underside of that hanging band is exactly the sine curve, so the two meet with no gap and no overlap anywhere and the top edge comes out flat at 1. The flat line is not drawn in at the end. It is what the two curves add up to.
Figure 6.1, first bar
The identity as one bar
sin²θ + cos²θ
= 1.000
- sin²θ 0.750
- cos²θ 0.250
Each bar is drawn to its own total, printed at its right end, so a bar whose total is in the millions still splits correctly. Teal is sin², amber cos², purple tan², and a dashed purple block is cot², the reciprocal of tan. A grey block is a plain 1.
sin²θ
cos²θ
Now the amber one is lifted and dropped onto the teal one.
The amber shape flips over and drops in. Its curved edge lands on sin²θ and its old baseline becomes the flat top at 1.
At 60°: sin²θ = 0.750, cos²θ = 0.250, total = 1.000.
The two panels at the top are the ingredients, one each, on their own axes: teal is sine, amber is cosine. Step 2 squares both, which folds the parts below the axis upwards and leaves everything between 0 and 1. The wide plot underneath is where they are put together. Step 3 flips the whole amber shape over and drops it in. Reflecting it about the axis and lifting it by 1 sends every value v to 1 minus v, so its curved edge becomes 1 minus cos²θ, which is sin²θ, and lands exactly on the teal curve. Its flat baseline arrives at the top as the line at 1. That straight edge was there all along; it was the axis. Drag anywhere on the wide plot to move the angle.
Step 3 of 6
Divide it through
The other two Pythagorean identities are the first one divided through by cos² and by sin².
Start from sin²θ + cos²θ = 1 and divide every term by cos²θ:
sin²θ/cos²θ + cos²θ/cos²θ = 1/cos²θ
The first term is (sin θ / cos θ)², which is tan²θ. The second is 1. The right side is (1 / cos θ)², which is sec²θ. So:
tan²θ + 1 = sec²θ
Now go back to the original and divide every term by sin²θ instead:
sin²θ/sin²θ + cos²θ/sin²θ = 1/sin²θ
which gives:
1 + cot²θ = csc²θ
These are the second and third Pythagorean identities, and they are usually presented as three separate things to learn. They are one thing. There is only one Pythagorean identity, and the other two are it, rescaled.
The geometry says the same thing. Dividing every side of the unit triangle by cos θ scales it up until the horizontal side is exactly 1, and that is precisely the large triangle from page 3 with sides 1, tan θ and sec θ. Dividing by sin θ instead scales it until the vertical side is 1, giving the triangle with sides cot θ, 1 and csc θ. Three identities, three scalings of one shape.
One honest caveat. Dividing by cos²θ is not allowed when cos θ is 0, so tan²θ + 1 = sec²θ holds everywhere except at 90 degrees and 270 degrees, where both sides are undefined anyway. The same applies to the third identity where sin θ is 0.
How to read this
Three versions of the same triangle side by side, each a scaled copy of the first. The first has hypotenuse 1, the second has its horizontal side scaled to 1, the third has its vertical side scaled to 1. Both arrows measure from the first triangle rather than from the one beside them: each reads "from A" and then names its factor, 1/cos θ for the second and 1/sin θ for the third, followed by the number that factor currently comes to. All three carry the same angle θ, which is what makes them similar triangles, and the identity written under each is Pythagoras applied to that particular scaling. Scaling by 1/cos θ runs away as cos θ approaches 0, so the drawing shrinks to fit and prints the factor it shrank by. The three bars underneath are the three identities as arithmetic, each drawn to its own total.
Figure 6.1
One triangle, at three scales
A the unit circle triangle
- opposite sin θ 0.866
- adjacent cos θ 0.500
- hypotenuse 1 1.000
B A with every side divided by cos θ
- opposite tan θ 1.732
- adjacent 1 1.000
- hypotenuse sec θ 2.000
C A with every side divided by sin θ
- opposite 1 1.000
- adjacent cot θ 0.577
- hypotenuse csc θ 1.155
Drawn at true size: all three fit as they are.
Each side is drawn in the colour of the quantity it equals, not the job it does. Teal is the sine family, amber the cosine family, purple the tangent family. A solid side is the function itself and a dashed one its reciprocal, so sec is dashed amber because it belongs to cos. A black side is a plain 1. The three triangles are the same shape at three sizes, which is why all three carry the same angle θ.
Figure 6.1, continued
The three identities as bars
sin²θ + cos²θ
= 1.000
- sin²θ 0.750
- cos²θ 0.250
tan²θ + 1
= 4.000 = sec²θ
- tan²θ 3.000
- 1 1.000
1 + cot²θ
= 1.333 = csc²θ
- 1 1.000
- cot²θ 0.333
Each bar is drawn to its own total, printed at its right end, so a bar whose total is in the millions still splits correctly. Teal is sin², amber cos², purple tan², and a dashed purple block is cot², the reciprocal of tan. A grey block is a plain 1.
Step 4 of 6
The map
The reciprocal and quotient identities connect all six functions into one map.
Collect everything that links the six.
The reciprocal identities: sec θ = 1/cos θ, csc θ = 1/sin θ, cot θ = 1/tan θ. Each pair multiplies to 1.
The quotient identities: tan θ = sin θ / cos θ, and cot θ = cos θ / sin θ.
The cofunction identities: cos θ = sin(90 degrees minus θ), cot θ = tan(90 degrees minus θ), csc θ = sec(90 degrees minus θ). Each function’s "co" partner is the same function of the complementary angle.
The Pythagorean identities: sin²θ + cos²θ = 1, tan²θ + 1 = sec²θ, 1 + cot²θ = csc²θ.
Put together, these mean the six functions carry very little independent information. Given sine and cosine you can write all six. Given sine alone and the quadrant, you can recover cosine from the first identity and then everything else. The set of six exists because different problems make different ones convenient, not because there are six separate ideas.
Historical footnote: navigation and surveying tables listed all six because a table lookup was cheaper than a division by hand. Secant and cosecant have largely fallen out of use for that reason, since a calculator makes 1 divided by cosine no harder than cosine.
How to read this
The six functions as chips on a hexagon, with the number 1 at its centre. Three link styles carry the three kinds of relation, and the legend names each one. Opposite chips are reciprocals and multiply to 1. Any chip is the product of the two chips beside it, which is where the quotient identities live. A chip and its mirror image across the horizontal middle are cofunctions. Pick any chip and the arithmetic for all three of its links is printed underneath at the current angle, including the angles where a link says nothing because one of its values is undefined.
Figure 6.2
Every relation on one hexagon
sin sits upper left. Its three links at this angle:
The three shaded triangles are the three Pythagorean identities
Selecting one also outlines its triangle in figure 6.1.
sin 0.866 · cos 0.500 · tan 1.732 · csc 1.155 · sec 2.000 · cot 0.577
A line through the centre joins a function to its reciprocal, and the pair multiplies to 1. An edge of the hexagon joins a chip to a neighbour, and every chip is the product of its two neighbours, which is where the quotient identities live. A dashed vertical line joins a chip to its mirror image across the middle, which is its cofunction. The line between tan and cot is drawn twice, one copy shifted sideways, because cot is both the reciprocal of tan and its cofunction. Colour sits on the underline of each chip, not on the word: teal is the sine family, amber the cosine family, purple the tangent family, and a dashed underline marks a reciprocal. Values are rounded to three decimals, so the printed factors may not multiply out to the last digit of the printed result.
Step 5 of 6
Where the exact values come from
The exact values at 0, 30, 45, 60 and 90 degrees come from cutting up two familiar shapes.
Two constructions produce every exact value anyone asks you to know.
Take a square with sides of length 1 and cut it corner to corner. The result is a right triangle with two legs of 1 and a hypotenuse of √2, by Pythagoras, since 1² + 1² = 2. Its two acute angles are equal and add to 90 degrees, so both are 45 degrees. Therefore sin(45 degrees) = cos(45 degrees) = 1/√2, usually written √2/2, which is 0.707 to three decimal places. And tan(45 degrees) = 1 exactly.
Take an equilateral triangle with sides of length 2, all angles 60 degrees, and cut it down the middle from a corner to the midpoint of the opposite side. Each half is a right triangle with a hypotenuse of 2, a short side of 1, and a third side of √3, since 2² minus 1² = 3. Its angles are 30, 60 and 90 degrees. Therefore sin(30 degrees) = 1/2 exactly, cos(30 degrees) = √3/2 which is 0.866 to three decimal places, sin(60 degrees) = √3/2, and cos(60 degrees) = 1/2.
The values at 0 and 90 degrees come from the circle rather than a triangle. At 0 the point is at (1, 0), so cos is 1 and sin is 0. At 90 degrees the point is at (0, 1), so cos is 0 and sin is 1. Tangent is 0 at 0 degrees and undefined at 90 degrees, since it divides by a cosine of 0.
A is a root left in exact form because it cannot be written as a fraction of whole numbers, such as √2 or √3. Keeping values as surds keeps them exact. √2/2 is precise; 0.707 is a rounded approximation, correct to three decimal places and wrong after that. Both are useful, and it is worth knowing which one you are holding.
The full table, exact then rounded to three decimal places where the exact form is not already a whole number or a half:
- 0 degrees (0 radians): sin 0, cos 1, tan 0, sec 1, csc undefined, cot undefined.
- 30 degrees (π/6): sin 1/2, cos √3/2 (0.866), tan 1/√3 (0.577), sec 2/√3 (1.155), csc 2, cot √3 (1.732).
- 45 degrees (π/4): sin √2/2 (0.707), cos √2/2 (0.707), tan 1, sec √2 (1.414), csc √2 (1.414), cot 1.
- 60 degrees (π/3): sin √3/2 (0.866), cos 1/2, tan √3 (1.732), sec 2, csc 2/√3 (1.155), cot 1/√3 (0.577).
- 90 degrees (π/2): sin 1, cos 0, tan undefined, sec undefined, csc 1, cot 0.
One pattern worth noticing in the sine column: 0, 1/2, √2/2, √3/2, 1 can be written as √0/2, √1/2, √2/2, √3/2, √4/2. The cosine column is the same list backwards, which is the cofunction relationship showing up in a table.
How to read this
The square and the equilateral triangle, each shown whole and then cut, with the side lengths marked, including the √2 and the √3 that Pythagoras supplies. Then the table of all six functions at the five angles, in exact form with the decimal underneath. Selecting a row sets the angle for the whole page, so every other figure moves to the triangle that row describes. The undefined entries are marked as undefined rather than left blank, because a missing value and a value of zero are different things. The staircase at the end is the sine column drawn as five heights, with the cosine column beneath it as the same five heights in the opposite order.
Figure 6.3
Where 30, 45 and 60 come from
A square, cut corner to corner
Both legs are 1, so the diagonal is √2 by Pythagoras, since 1² + 1² = 2. The two acute angles are equal and add to 90 degrees, so both are 45 degrees.
An equilateral triangle, cut down the middle
Every side is 2 and every angle 60 degrees. The cut halves the base into 1 and 1, and the third side is √3 by Pythagoras, since 2² minus 1² = 3. The half kept has angles 30, 60 and 90 degrees.
Neither shape wears a function colour: these are constructions, not functions. The outline and the cut are black, the measurements grey, and the half that gets thrown away is faded to 20%. Every length here is exact: 1, 2, √2 and √3, with the decimals printed to five places.
Figure 6.4
The exact values
| degrees | radians | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|---|
| 0 | 0 0.0000 | 1 1.0000 | 0 0.0000 | undefined undefined | 1 1.0000 | undefined undefined | |
| π/6 | 1/2 0.5000 | √3/2 0.8660 | 1/√3 0.5774 | 2 2.0000 | 2/√3 1.1547 | √3 1.7321 | |
| π/4 | √2/2 0.7071 | √2/2 0.7071 | 1 1.0000 | √2 1.4142 | √2 1.4142 | 1 1.0000 | |
| π/3 | √3/2 0.8660 | 1/2 0.5000 | √3 1.7321 | 2/√3 1.1547 | 2 2.0000 | 1/√3 0.5774 | |
| π/2 | 1 1.0000 | 0 0.0000 | undefined undefined | 1 1.0000 | undefined undefined | 0 0.0000 |
A solid swatch marks a function, a dashed one its reciprocal, and the hue marks the family: teal for sine, amber for cosine, purple for tangent. The exact form is on top and the value to four decimals underneath. A cell that reads undefined is undefined, which is a different thing from zero. Selecting a row sets the angle for every figure on this page.
Figure 6.5
The staircase
Teal: sin, heights √n/2 for n = 0, 1, 2, 3, 4. Amber: cos, the same five heights in the opposite order. Two rows, one pattern, read in both directions. The number under each column is that n, and selecting an angle sets it for every figure on this page.
Step 6 of 6
Where this gets used
Where these six functions are actually used.
Waves and signals. Any repeating signal can be written as a sum of sines and cosines at different frequencies, which is Fourier’s result from 1822. That decomposition is the basis of audio compression, image compression, radio, and most of digital signal processing. When an equaliser shows you frequency bands, it is showing you the sizes of those sine components.
Circular and rotational motion. A point on a spinning wheel has position (r cos θ, r sin θ) at angle θ. Anything that orbits, spins, oscillates or swings gets described this way, from a pendulum to a planet to alternating current, which is called alternating because its voltage follows a sine curve.
Surveying and navigation. Measure one distance and two angles and trigonometry gives you every other distance in the triangle, which is how land was mapped before satellites and how satellite positioning works now. The historic tables of all six functions existed for this work.
Computer graphics. Rotating a point by angle θ about the origin sends (x, y) to (x cos θ minus y sin θ, x sin θ + y cos θ). Every rotation on a screen, in a game engine or in a 3D modeller, runs that calculation or a close relative of it. The same pair of numbers that were the coordinates of a point on the unit circle now serve as the entries of a rotation matrix.
The common thread is that all four are about things that turn or repeat. That is what the circle definition on page 2 bought, and it is why the triangle definition on page 1 was not enough.
How to read this
One of those four is built here, the rotation. An L shaped outline is drawn twice: pale where it starts, solid where the rotation sends it. Beside it is the 2 by 2 matrix doing the work, with the two cos θ entries marked in the cosine colour and the two sin θ entries in the sine colour, carrying the values at the current angle. The other three applications are described above and are not drawn. Nothing new feeds this figure: the two numbers in the matrix are the same sine and cosine as everywhere else on the page.
Figure 6.6
The same two numbers, rotating a shape
at θ = 60°: cos θ = 0.500, sin θ = 0.866
The corner at (1.000, −1.500) goes to (1.799, 0.116), which is (x cos θ minus y sin θ, x sin θ + y cos θ) worked out with those two numbers.
Amber marks the two cos θ entries of the matrix and teal the two sin θ entries, the same two colours those functions wear everywhere else on this page. The pale outline is where the shape starts and the solid one is where the rotation sends it. Nothing new feeds this figure: it is the sine and cosine of the same angle, doing a different job.