Tiō chhin-chhiūⁿ (cos t, sin t) chia-ê tiám hêng-sêng chi̍t ê tan-ūi-îⁿ kāng-khoán, (cosh t, sinh t) chia-ê tiám ē tan-ūi siang-khiok-sòaⁿ ê chiàⁿ-chhiú pòaⁿ-pêng. Koh ū chi̍t ê sio-siâng ê só͘-chāi sī, sin(t) kap cos(t) ê tō-hâm-sò͘ hun-pia̍t sī cos(t) kap –sin(t), ah sinh(t) kap cosh(t) ê tō-hâm-só͘ hun-pia̍t sī cosh(t) kap +sinh(t).
Siang-khiok hâm-sò͘ chhut-hiān tī siang-khiok kì-hō-ha̍k tiong kak-tō͘ kap kī-lī ê kè-sǹg. They also occur in the solutions of many sòaⁿ-sèng bî-hun hong-têng-sek (phì-lūn-kóng tēng-gī catenary ê hong-têng-sek), li̍p-hong hong-têng-sek, kap Khu-chhioh-kak chō-piau tiong ê Laplace hong-têng-sek. Laplace hong-têng-sek tī bu̍t-lí-ha̍k ê chin-chē léng-he̍k tiong lóng chiok tiōng-iàu ·ê, chia-ê léng-he̍k pau-koat tiān-chú-ha̍k, jia̍t thoân-tō, liû-thé le̍k-ha̍k kap te̍k-sû siong-tùi-lūn.
Kā chí-sò͘ hâm-sò͘ hun-kái chò i ê khia-sò͘ pō͘-hūn kap siang-sò͘ pō͘-hūn, lán tiō ū hêng-téng-sek
kap
Kap Euler kong-sek ha̍p-pèng
tiō ē tit-tio̍h
che tiō sī it-poaⁿ ê ho̍k-cha̍p-sò͘ chí-sò͘ hâm-sò͘ ê kong-sek.
Lēng-gōa koh ū
Ho̍k-cha̍p-sò͘ piān-só͘ ê siang-khiok hâm-sò͘
In-ūi chí-sò͘ hâm-sò͘ ē-sái tùi jīm-hô ho̍k-cha̍p-sò͘ piān-só͘ lâi tēng-gī, lán tiō ē-sái kā hit khoán tēng-gī chhun kàu siang-khiok hâm-sò͘.
Ho̍k-cha̍p-sò͘ ê Euler kong-sek
só͘-í:
Só͘-í, siang-khiok hâm-sò͘ tùi in ê hi-pō͘ lâi kóng, sī chiu-kî hâm-sò͘, chiu-kî sī (iah-m̄-koh siang-khiok tangent kap cotangent ê chiu-kî sī ).
Ho̍k-pêⁿ-bīn téng ê Siang-khiok hâm-sò͘
Chham-khó chu-liāu
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