This rule is fairly unique in the manner it calculates trig and hyperbolic functions.
(Flip to the back side of the rule; the tour starts there.)

Scale Function
P/Q/Q′ These scales compute the Pythagorean theorem \(c=\sqrt{a^2+b^2}\). Q′ is an extension of Q.  Examples
\(\theta\) Argument to trig functions in degrees. Read \(sin\,\theta\) on P scale. Read \(tan\,\theta\) on T scale on front side of rule. Read \(cos\,\theta\) by setting left index of Q opposite \(sin\,\theta\) on P scale and reading \(cos\,\theta\) opposite right index of P. [This computes \(cos\,\theta=\sqrt{1-sin^2\theta}\).]  Examples
\(R_\theta\) Argument to trig functions in radians. Can be used with \(\theta\) scale to convert between degrees and radians.
\(G_\theta\) Argument to the Gudermannian (gd) function. Value of \(gd(x)\) is read on the \(R_\theta\) scale. Here's the magic to the Gudermannian function:
     \(sinh\,x=tan(gd(x))\)
     \(tanh\,x=sin(gd(x))\)
     \(cosh\,x=sec(gd(x))\)
So by setting the cursor to \(x\) on the \(G_\theta\) scale, you read \(sinh\,x\) on the T scale and \(tanh\,x\) on the P scale. Read \(cosh\,x\) by setting the left index of Q under the hairline and reading \(sech\,x\) opposite the right index of P, flip the rule over and use C/CI to invert. If \(sinh\,x\le 1\), there is another way to compute \(cosh\,x\): set the value of \(sinh\,x\) on Q under the hairline and read \(cosh\,x\) on Q′. [The first method computes \(sech\,\theta=\sqrt{1-tanh^2\theta},\) \(cosh\,x=1/sech\,x\); the second computes \(cosh\,x=\sqrt{1+sinh^2x}\).]  Examples

There are some links to documentation on this rule's main page.

Read more about the Gudermannian function on Wikipedia.

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