Trigonometric Identities and Triangles
The Addition, Double-Angle and Half-Angle Formulae
The two addition formulae, everything derived from them, and how each derivation works — subtraction, tangent, double angles and half angles in turn.
What this page covers
- State the addition formulae for sine and cosine
- Derive the subtraction formulae using the parity identities
- Derive the double-angle formulae by setting the two angles equal
- Use half-angle and product-to-sum formulae
The two addition formulae
Sine carries a plus; cosine carries a minus. This asymmetry is the most common source of error in the whole topic. A quick check: at u = v = π2, cos(π) = -1, and cosπ2cosπ2 - sinπ2sinπ2 = 0 - 1 = -1 ✓. With a plus it would give +1, which is wrong.
The source establishes the formulae geometrically, with a diagram of stacked right triangles inside the unit circle whose sides are labelled cos ucos v, sin usin v and so on. The labelled projections are the four products in the formulae.
Worked example — the source's use
Find sin5pi12 exactly.
Split the angle into two whose values are known:
√3 + 12√2 ≈ 2.7322.828 ≈ 0.966, and 5pi12 = 75° with sin 75° ≈ 0.966 ✓.
The subtraction formulae
The source derives these rather than stating them, by replacing v with -v and using the parity identities.
Sine
Cosine
For sine the sign follows the one in the argument; for cosine it is opposite. So sin(u - v) takes a minus and cos(u - v) takes a plus.
A first consequence
The source immediately applies the formula to derive the complementary-angle relation.
This is the origin of the prefix co: the cosine of an angle is the sine of its complement. The same relation connects tangent with cotangent and secant with cosecant.
The tangent formula
Dividing the sine formula by the cosine formula, then dividing top and bottom by cos ucos v, gives the tangent version. The source does both steps in full.
Dividing every term by cos u cos v turns each ratio into a tangent:
The source notices a consequence worth keeping: if u - v = π2 then the denominator 1 + tan utan v must vanish, so tan u tan v = -1. That is the perpendicular-slope condition m1m2 = -1 arriving from a different direction — a line's slope is the tangent of its angle of inclination.
Double angles
Setting u = v = θ in the addition formulae gives the double-angle results immediately.
Sine
Cosine, in three equivalent forms
Substituting sin2θ = 1 - cos2θ:
Substituting cos2θ = 1 - sin2θ instead:
All three are the same identity. Choose whichever leaves the expression in the function you want to keep.
Worked example — the source's check
Verify sinπ3 using the double-angle formula with θ = π6.
Half angles
Rearranging cos 2theta = 1 - 2sin2θ isolates sin2θ, and putting θ = a2 turns it into a half-angle formula. The source does exactly this.
It arises from taking a square root, and the formula cannot resolve it. The sign is fixed by which quadrant a2 lies in, determined by the ASTC rule. Writing the formula without the ± asserts a sign that has not been established.
Worked example — the source's case
Find sinπ12 using a = π6.
√34 ≈ 0.433, so the radicand is about 0.067 and the root about 0.259. Since π12 = 15° and sin 15° ≈ 0.259 ✓.
Product to sum
Adding the sine addition and subtraction formulae makes the cos usin v terms cancel, leaving a product expressed as a sum.
Subtracting instead of adding, and doing the same with the cosine formulae, yields the other three of the family. The practical importance is in signal work, where a product of two waves is exactly a sum of two others — which is how mixing produces sum and difference frequencies.
Summary
| Formula | Derived from |
|---|---|
| sin(u + v) = sin ucos v + cos usin v | Geometry |
| cos(u + v) = cos ucos v - sin usin v | Geometry |
| sin(u - v) = sin ucos v - cos usin v | Replace v by -v |
| cos(u - v) = cos ucos v + sin usin v | Replace v by -v |
| tan(u + v) = tan u + tan v1 - tan utan v | Divide sine by cosine |
| sin 2theta = 2sin θcos θ | Set u = v |
| cos 2theta = 2cos2θ - 1 | Set u = v, then Pythagoras |
| sina2 = ±√1 - cos a2 | Rearrange the double angle |
| sin ucos v = 12[sin(u+v) + sin(u-v)] | Add the two sine formulae |
Two formulae at the top; everything else derived. That is the efficient way to hold this material, and it is how the source presents it.
Common mistakes
| Mistake | Correct | Check |
|---|---|---|
| cos(u + v) = cos ucos v + sin usin v | The sign is minus | At u = v = π2 it must give -1 |
| sin(u + v) = sin u + sin v | Not a valid step | At u = v = π2: 0 against 2 |
| cos 2theta = 2cos θ | = 2cos2θ - 1 | At θ = 0: 1, not 2 |
| Dropping the ± in a half-angle formula | The quadrant decides | It comes from a square root |
| tan(u + v) = tan u + tan v | There is a denominator | Derive it from the ratio |
| Using sin 2theta = sin θ cos θ | The factor 2 is required | At θ = π4: 1, not 12 |
Frequently asked questions
Why is the cosine formula's sign reversed?
cos(u + v) = cos ucos v - sin usin v carries a minus while the sine formula carries a plus. It is the single most-forgotten detail in the topic. Checking at u = v = 0 does not distinguish them; checking at u = v = π2 does.
Do I need to memorise all of these?
Only the two addition formulae. Every other formula on this page is one or two lines of derivation from them, and the source derives each one rather than asserting it.
Which double-angle form for cosine should I use?
Whichever leaves the expression in the function you want. 2cos2θ - 1 if you want cosine, 1 - 2sin2θ if you want sine.
Why does the half-angle formula carry a ±?
Because it comes from taking a square root. The sign is decided by which quadrant a2 lies in, not by the formula.
Source. Handwritten teaching notes, Week 9, pages 2-6.
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.