Trigonometry, starting from what it is actually for
Begin with the question the whole subject was invented to answer — how tall is that thing you cannot climb? — then work up to the full table of sin, cos, tan, cosec, sec and cot for every angle from 0° to 360°.
So what is trigonometry actually for?
Before a single formula. It is one practical idea, worked out by people who had to measure land, seas and stars they could not walk up to.
The problem it solves
Measuring what you cannot reach.
- The height of a tower
- The width of a river
- How steep a hill is
- Where a satellite sits
What it needs from you
Two measurements, taken standing still.
- One angle — how far you tilt your head
- One length you can walk along
- Usually just the flat ground
What it hands back
Every other length in that triangle.
- You give it an angle
- It gives you back a number
- That number scales what you measured into what you wanted
Start here: how tall is that tower?
You cannot climb it and you cannot run a tape measure up it. But you can measure two easy things on the ground — and that is the entire reason trigonometry was invented.
Watch the numbers as it moves. The distance changes. The angle changes. The height never does. That fixed relationship between an angle and the sides of its triangle is all a trigonometric ratio is — and the table further down is just the answers written out in advance.
Every side gets a name, and the names do the work
Pick a corner to stand in — that is θ. The three sides are then named from where you are standing, and each ratio is simply two of them divided.
Why one table can answer for every triangle
Here is the idea the whole subject rests on. Keep the angle at 35° and make the triangle as big or as small as you like — every side changes, and the division between any two of them does not budge.
Watch the middle column: the divisions on the left keep changing, and they keep landing on the same three answers. A ratio belongs to the angle, not to the triangle. So somebody worked them out once, wrote them down, and nobody has had to do it again since. That list is the trigonometry table.
Now watch one get solved, line by line
Your school flagpole. You stand 30 m back and measure the angle up to the top as 30°. How tall is it? Six lines, and not one of them is harder than a multiplication.
- 1Write down what you have, and what you want
distance = 30 m, angle = 30°, height = ?You paced 30 m back from the pole, and measured the angle up to its top as 30°. The height is the thing you do not have. - 2Name the two sides that matter
opposite = height (unknown), adjacent = 30 m - 3Pick the ratio built from those two
tan θ = opposite ÷ adjacent - 4Put your numbers in
tan 30° = height ÷ 30 - 5Look up what tan 30° is worth
tan 30° = 1/√3 = 0.577 - 6Rearrange, and multiply
height = 30 × 0.577 = 17.3 m
Every trigonometry question you will be set is this shape: you know two things about a right triangle and you want a third. Name the sides, pick the ratio, look up the number, multiply. Keep pressing Next and watch where the number in step 5 comes from.
Once you can see it, it is everywhere
The tower was the classroom version. Here is the same trick running in four places you already meet — pick one and watch the number move.
Ramps, roofs and stairs
Every ramp and roof is specified by its steepness, and steepness is a tangent. A wheelchair ramp is not allowed to be steeper than 1 in 12 — that rule is tan θ ≤ 0.083.
A slope of 1 in 5.0. The legal limit for a ramp is 1 in 12, which is tan θ = 0.083 — so the moment this number climbs past 0.083, the ramp stops being usable.
Now spin that triangle all the way round
One triangle answers one angle. Put its corner at the centre of a circle and sweep it, and you get the answer for every angle at once — which is exactly what the table is a list of.
Add Sugar To Coffee. Quadrant by quadrant, that is which ratios stay positive — everything else picks up a minus sign.
The table itself
Tap any value and it will show you the triangle it came out of.
| Angle | sin θopposite ÷ hypotenuse | cos θadjacent ÷ hypotenuse | tan θopposite ÷ adjacent | cosec θhypotenuse ÷ opposite | sec θhypotenuse ÷ adjacent | cot θadjacent ÷ opposite |
|---|---|---|---|---|---|---|
| 0°0 | ||||||
| 30°π/6 | ||||||
| 45°π/4 | ||||||
| 60°π/3 | ||||||
| 90°π/2 |
ND means not defined — at that angle the ratio is asking you to divide by zero.
The sine row, from counting to four
There is nothing to memorise here. Step through it once and you can rebuild the row in the margin of an exam paper.
The six worth carrying with you
Most questions that look impossible are one of these wearing a disguise.