11 Functions of Sums and Diﬀerences.

Article 11
Functions of Sums and Diﬀerences.

(a) To prove the diﬀerence-formulas

\begin{aligned} sinh (u - v) & = sinh u cosh v - cosh u sinh v, \\ cosh (u - v) & = cosh u cosh v - sinh u sinh v . \end{aligned}\}

(19)

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Let $O A$ be any radius of a hyperbola, and let the sectors $A O P, A O Q$ have the measures $u, v$ ; then $u - v$ is the measure of the sector $Q O P$ . Let $O B, O Q'$ be the radii conjugate to $O A, O Q$ ; and let the coördinates of $P, Q, Q'$ be $(x_{1}, y_{1})$ , $(x, y)$ , $(x', y')$ with reference to the axes $O A, O B$ ; then

\begin{array}{l} sinh (u - v) & = sinh \frac{sector Q O P}{K} = \frac{triangle Q O P}{K} & [Art. 5. \\ = \frac{\frac{1}{2} (x y_{1} - x_{1} y) sin ω}{\frac{1}{2} a_{1} b_{1} sin ω} = \frac{y_{1}}{b_{1}} \cdot \frac{x}{a_{1}} - \frac{y}{b_{1}} \cdot \frac{x_{1}}{a_{1}} \\ = sinh u cosh v - cosh u sinh v; \end{array}

\begin{array}{l} cosh (u - v) & = cosh \frac{sector Q O P}{K} = \frac{triangle P O Q'}{K} & [Art. 5. \\ = \frac{\frac{1}{2} (x_{1} y' - y_{1} x') sin ω}{\frac{1}{2} a_{1} b_{1} sin ω} = \frac{y'}{b_{1}} \cdot \frac{x_{1}}{a_{1}} - \frac{y}{b_{1}} \cdot \frac{x'}{a_{1}}; \\ but \\ \frac{y'}{b_{1}} & = \frac{x}{a_{1}}, \frac{x'}{a_{1}} = \frac{y}{b_{1}}, & (20) \end{array}

since

Q, Q'

are extremities of conjugate radii; hence

cosh (u - v) = cosh u cosh v - sinh u sinh v .

In the ﬁgures $u$ is positive and $v$ is positive or negative. Other ﬁgures may be drawn with $u$ negative, and the language in the text will apply to all. In the case of elliptic sectors, similar ﬁgures may be drawn, and the same language will apply, except that the second equation of (20) will be $\frac{x'}{a_{1}} = \frac{- y}{b_{1}}$ ; therefore

\begin{array}{l} sin (u - v) & = sin u cos v - cos u sin v, \\ cos (u - v) & = cos u cos v + sin u sin v . \end{array}

(b) To prove the sum-formulas

\begin{aligned} sinh (u + v) & = sinh u cosh v + cosh u sinh v, \\ cosh (u + v) & = cosh u cosh v + sinh u sinh v . \end{aligned}\}

(21)

These equations follow from (19) by changing $v$ into $- v$ , and then for $sinh (- v)$ , $cosh (- v)$ , writing $- sinh v$ , $cosh v$ (Art. 9, eqs. (18)).

tanh (u \pm v) = \frac{tanh u \pm tanh v}{1 \pm tanh u tanh v} .

(22)

Writing $tanh (u \pm v) = \frac{sinh (u \pm v)}{cosh (u \pm v)}$ , expanding and dividing numerator and denominator by $cosh u cosh v$ , eq. (22) is obtained.

Prob. 16.

Given

cosh u = 2, cosh v = 3

, ﬁnd

cosh (u + v)

Prob. 17.

Prove the following identities:

(a): $sinh 2 u = 2 sinh u cosh u$ .
(b): $cosh 2 u = {cosh}^{2} u + {sinh}^{2} u = 1 + 2 {sinh}^{2} u = 2 {cosh}^{2} u - 1$ .
(c): $1 + cosh u = 2 {cosh}^{2} \frac{1}{2} u, cosh u - 1 = 2 {sinh}^{2} \frac{1}{2} u$ .
(d): $tanh \frac{1}{2} u = \frac{sinh u}{1 + cosh u} = \frac{cosh u - 1}{sinh u} = {(\frac{cosh u - 1}{cosh u + 1})}^{\frac{1}{2}}$ .
(e): $sinh 2 u = \frac{2 tanh u}{1 - {tanh}^{2} u}, cosh 2 u = \frac{1 + {tanh}^{2} u}{1 - {tanh}^{2} u}$ .
(f): $sinh 3 u = 3 sinh u + 4 {sinh}^{3} u, cosh 3 u = 4 {cosh}^{3} u - 3 cosh u$ .
(g): $cosh u + sinh u = \frac{1 + tanh \frac{1}{2} u}{1 - tanh \frac{1}{2} u}$ .
(h): $(cosh u + sinh u) (cosh v + sinh v) = cosh (u + v) + sinh (u + v)$ .
(i): Generalize (h); and show also what it becomes when $u = v = \dots$
(j): ${sinh}^{2} x {cos}^{2} y + {cosh}^{2} x {sin}^{2} y = {sinh}^{2} x + {sin}^{2} y$ .
(k): ${cosh}^{- 1} m \pm {cosh}^{- 1} n = {cosh}^{- 1} [m n \pm \sqrt{(m^{2} - 1) (n^{2} - 1)}]$ .
(l): ${sinh}^{- 1} m \pm {sinh}^{- 1} n = {sinh}^{- 1} [m \sqrt{1 + n^{2}} \pm n \sqrt{1 + m^{2}}]$ .

Prob. 18.

What modiﬁcations of signs are required in (21), (22), in order to pass to circular functions?

Prob. 19.

Modify the identities of Prob. 17 for the same purpose.

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Article 11Functions of Sums and Diﬀerences.

Article 11
Functions of Sums and Diﬀerences.