Osborn's Rule, and Where It Fails
Chapter Thirty-Two
Syllabus topic Module 1, "1.1 Complex Numbers"
Pages 72 to 73 of 303
In one line
Take any circular identity, change every function to its hyperbolic namesake, and change the sign of any term containing a product of two sines.
The rule
Osborn's rule is a conversion recipe, and it works because of the relations in the previous chapter.
- Replace cos by cosh, sin by sinh, tan by tanh, and so on.
- Wherever a term contains the product of an even number of sines, other than none, change its sign. In practice: a term with two sinh factors, or a sinh squared, or a tan squared, flips sign.
That is the whole rule. It is a labour-saving device and not a proof, and a question asking you to "prove" a hyperbolic identity wants the exponential derivation of the previous chapters, not this.
Why it works
Consider a circular identity in which every angle is replaced by i times a real variable. By the relations, every cosine becomes a cosh with no i, and every sine becomes i times a sinh. So a term with two sine factors picks up i squared, which is minus one, and a term with no sine factors or one sine factor picks up either nothing or a single i that can be divided out of the whole identity.
Hence: an even number of sines, at least two, flips the sign. Everything else is unchanged. That is Osborn's rule, and it is the same argument every time.
Worked conversions
The Pythagorean identity. The circular one has two sines in the second term, which is sin squared, so its sign flips.
cos(x)^2 + sin(x)^2 = 1
cosh(x)^2 - sinh(x)^2 = 1
The compound-angle formula for the cosine. The term cos x cos y has no sines; the term sin x sin y has two, so its sign flips from minus to plus.
cos(x + y) = cos(x) cos(y) - sin(x) sin(y)
cosh(x + y) = cosh(x) cosh(y) + sinh(x) sinh(y)
The compound-angle formula for the sine. Each term has exactly one sine factor, so nothing flips.
sin(x + y) = sin(x) cos(y) + cos(x) sin(y)
sinh(x + y) = sinh(x) cosh(y) + cosh(x) sinh(y)
The double-angle formula for the cosine.
cos(2x) = cos(x)^2 - sin(x)^2
cosh(2x) = cosh(x)^2 + sinh(x)^2
The tangent formula. A tangent counts as a sine over a cosine, so tan x tan y counts as two sines and flips.
tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) tan(y))
tanh(x + y) = (tanh(x) + tanh(y))/(1 + tanh(x) tanh(y))
The triple-angle formula for the sine. The term sin cubed has three sines, an odd number, so it does not flip; but 3 sin x has one, so it does not either. And yet the hyperbolic version does differ in sign.
Osborn's Rule, and Where It Fails
sin(3x) = 3 sin(x) - 4 sin(x)^3
sinh(3x) = 3 sinh(x) + 4 sinh(x)^3
That looks like a counterexample and is not. Dividing the whole identity by i removes one factor of i from each side, and the sin cubed term is left with i squared, which is the flip. The rule as stated in step 2 is about an even number of sines after that division has been accounted for, and the practical way to say it is: count the sine factors, and flip the sign when the count is even, counting the overall factor of i as one of them. For an identity whose left-hand side is a sine, that means a term with an odd number of sines flips.
That subtlety is why the rule is a shortcut and not a substitute for the derivation. The honest advice: use Osborn's rule to guess, then check the guess from the definitions. A wrong sign in an identity is worth no marks at all, and it takes three lines to confirm.
Where it fails outright
Osborn's rule says nothing about anything that is not an identity between products and sums of the functions.
- Ranges. The cosine is at most one; the cosh is at least one. No sign rule will tell you that.
- Derivatives. The derivative of cos is minus sin; the derivative of cosh is plus sinh. There is no sine product here, so the rule would predict no change, and yet the sign does change.
- Periods. The circular functions have period 2 pi; the hyperbolic ones have period 2 pi i.
- Zeros. The cosine vanishes at odd multiples of pi over two; the cosh has no real zero at all.
So the rule converts algebraic identities and nothing else.
Check yourself
Convert each circular identity with the rule, then confirm against the hyperbolic chapter.
| Circular | Hyperbolic |
|---|---|
| cos squared plus sin squared is one | cosh squared minus sinh squared is one |
| one plus tan squared is sec squared | one minus tanh squared is sech squared |
| cos 2x as 1 minus 2 sin squared | cosh 2x as 1 plus 2 sinh squared |
| sin 2x as twice sin cos | sinh 2x as twice sinh cosh |
And the machine-checked versions of the last two:
cosh(2x) = 1 + 2 sinh(x)^2
sinh(2x) = 2 sinh(x) cosh(x)
1 - tanh(x)^2 = sech(x)^2
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself for the same subject.