Τριγωνομετρικές συναρτήσεις: ημίτονο της γωνίας
Γιατί χρειάζεται να γνωρίζετε την τιμή του ημιτόνου? Φανταστείτε μια κατάσταση: μία από τις γωνίες (Α=60⁰) εγγεγραμμένη σε ένα ορθογώνιο τρίγωνο και το μήκος της υποτείνουσας είναι γνωστό. Δεν υπάρχει άλλη πληροφορία. Είναι απαραίτητο να υπολογιστεί η απέναντι πλευρά προς τη γωνία (Α) πλευρά. Πώς να προχωρήσετε?
Η κατάσταση είναι πολύ απλή: κοιτάξτε στους πίνακες Μπράντις, βρείτε την τιμή sin(60⁰)=0,866, υποκαταστήστε τα δεδομένα στον τύπο της τριγωνομετρικής συνάρτησης και λύστε τη γραμμική εξίσωση. Από το σχολικό μάθημα, είναι γνωστό ότι sin η γωνία – είναι η αναλογία της απέναντι πλευράς προς τη γωνία, στην περίπτωση αυτή Α=60⁰, της πλευράς προς την υποτείνουσα.
Είναι πιο απλό να εκτελέσετε όλους τους υπολογισμούς χρησιμοποιώντας τον online υπολογιστή στον ιστότοπο. Με αυτόν τον τρόπο, μπορείτε να υπολογίσετε το μήκος οποιασδήποτε πλευράς ενός ορθογωνίου τριγώνου. Γνωρίζοντας τη γωνία – σημαίνει ότι γνωρίζουμε sin αυτή τη γωνία. Αντιστρόφως, γνωρίζοντας sin – η εύρεση της γωνίας δεν είναι πρόβλημα.
Πίνακας ημιτόνου 0°- 360°
| Sin(1°) | 0.0175 | | Sin(2°) | 0.0349 | | Sin(3°) | 0.0523 | | Sin(4°) | 0.0698 | | Sin(5°) | 0.0872 | | Sin(6°) | 0.1045 | | Sin(7°) | 0.1219 | | Sin(8°) | 0.1392 | | Sin(9°) | 0.1564 | | Sin(10°) | 0.1736 | | Sin(11°) | 0.1908 | | Sin(12°) | 0.2079 | | Sin(13°) | 0.225 | | Sin(14°) | 0.2419 | | Sin(15°) | 0.2588 | | Sin(16°) | 0.2756 | | Sin(17°) | 0.2924 | | Sin(18°) | 0.309 | | Sin(19°) | 0.3256 | | Sin(20°) | 0.342 | | Sin(21°) | 0.3584 | | Sin(22°) | 0.3746 | | Sin(23°) | 0.3907 | | Sin(24°) | 0.4067 | | Sin(25°) | 0.4226 | | Sin(26°) | 0.4384 | | Sin(27°) | 0.454 | | Sin(28°) | 0.4695 | | Sin(29°) | 0.4848 | | Sin(30°) | 0.5 | | Sin(31°) | 0.515 | | Sin(32°) | 0.5299 | | Sin(33°) | 0.5446 | | Sin(34°) | 0.5592 | | Sin(35°) | 0.5736 | | Sin(36°) | 0.5878 |
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| Sin(37°) | 0.6018 | | Sin(38°) | 0.6157 | | Sin(39°) | 0.6293 | | Sin(40°) | 0.6428 | | Sin(41°) | 0.6561 | | Sin(42°) | 0.6691 | | Sin(43°) | 0.682 | | Sin(44°) | 0.6947 | | Sin(45°) | 0.7071 | | Sin(46°) | 0.7193 | | Sin(47°) | 0.7314 | | Sin(48°) | 0.7431 | | Sin(49°) | 0.7547 | | Sin(50°) | 0.766 | | Sin(51°) | 0.7771 | | Sin(52°) | 0.788 | | Sin(53°) | 0.7986 | | Sin(54°) | 0.809 | | Sin(55°) | 0.8192 | | Sin(56°) | 0.829 | | Sin(57°) | 0.8387 | | Sin(58°) | 0.848 | | Sin(59°) | 0.8572 | | Sin(60°) | 0.866 | | Sin(61°) | 0.8746 | | Sin(62°) | 0.8829 | | Sin(63°) | 0.891 | | Sin(64°) | 0.8988 | | Sin(65°) | 0.9063 | | Sin(66°) | 0.9135 | | Sin(67°) | 0.9205 | | Sin(68°) | 0.9272 | | Sin(69°) | 0.9336 | | Sin(70°) | 0.9397 | | Sin(71°) | 0.9455 | | Sin(72°) | 0.9511 |
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| Sin(73°) | 0.9563 | | Sin(74°) | 0.9613 | | Sin(75°) | 0.9659 | | Sin(76°) | 0.9703 | | Sin(77°) | 0.9744 | | Sin(78°) | 0.9781 | | Sin(79°) | 0.9816 | | Sin(80°) | 0.9848 | | Sin(81°) | 0.9877 | | Sin(82°) | 0.9903 | | Sin(83°) | 0.9925 | | Sin(84°) | 0.9945 | | Sin(85°) | 0.9962 | | Sin(86°) | 0.9976 | | Sin(87°) | 0.9986 | | Sin(88°) | 0.9994 | | Sin(89°) | 0.9998 | | Sin(90°) | 1 | | Sin(91°) | 0.9998 | | Sin(92°) | 0.9994 | | Sin(93°) | 0.9986 | | Sin(94°) | 0.9976 | | Sin(95°) | 0.9962 | | Sin(96°) | 0.9945 | | Sin(97°) | 0.9925 | | Sin(98°) | 0.9903 | | Sin(99°) | 0.9877 | | Sin(100°) | 0.9848 | | Sin(101°) | 0.9816 | | Sin(102°) | 0.9781 | | Sin(103°) | 0.9744 | | Sin(104°) | 0.9703 | | Sin(105°) | 0.9659 | | Sin(106°) | 0.9613 | | Sin(107°) | 0.9563 | | Sin(108°) | 0.9511 |
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| Sin(109°) | 0.9455 | | Sin(110°) | 0.9397 | | Sin(111°) | 0.9336 | | Sin(112°) | 0.9272 | | Sin(113°) | 0.9205 | | Sin(114°) | 0.9135 | | Sin(115°) | 0.9063 | | Sin(116°) | 0.8988 | | Sin(117°) | 0.891 | | Sin(118°) | 0.8829 | | Sin(119°) | 0.8746 | | Sin(120°) | 0.866 | | Sin(121°) | 0.8572 | | Sin(122°) | 0.848 | | Sin(123°) | 0.8387 | | Sin(124°) | 0.829 | | Sin(125°) | 0.8192 | | Sin(126°) | 0.809 | | Sin(127°) | 0.7986 | | Sin(128°) | 0.788 | | Sin(129°) | 0.7771 | | Sin(130°) | 0.766 | | Sin(131°) | 0.7547 | | Sin(132°) | 0.7431 | | Sin(133°) | 0.7314 | | Sin(134°) | 0.7193 | | Sin(135°) | 0.7071 | | Sin(136°) | 0.6947 | | Sin(137°) | 0.682 | | Sin(138°) | 0.6691 | | Sin(139°) | 0.6561 | | Sin(140°) | 0.6428 | | Sin(141°) | 0.6293 | | Sin(142°) | 0.6157 | | Sin(143°) | 0.6018 | | Sin(144°) | 0.5878 |
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| Sin(145°) | 0.5736 | | Sin(146°) | 0.5592 | | Sin(147°) | 0.5446 | | Sin(148°) | 0.5299 | | Sin(149°) | 0.515 | | Sin(150°) | 0.5 | | Sin(151°) | 0.4848 | | Sin(152°) | 0.4695 | | Sin(153°) | 0.454 | | Sin(154°) | 0.4384 | | Sin(155°) | 0.4226 | | Sin(156°) | 0.4067 | | Sin(157°) | 0.3907 | | Sin(158°) | 0.3746 | | Sin(159°) | 0.3584 | | Sin(160°) | 0.342 | | Sin(161°) | 0.3256 | | Sin(162°) | 0.309 | | Sin(163°) | 0.2924 | | Sin(164°) | 0.2756 | | Sin(165°) | 0.2588 | | Sin(166°) | 0.2419 | | Sin(167°) | 0.225 | | Sin(168°) | 0.2079 | | Sin(169°) | 0.1908 | | Sin(170°) | 0.1736 | | Sin(171°) | 0.1564 | | Sin(172°) | 0.1392 | | Sin(173°) | 0.1219 | | Sin(174°) | 0.1045 | | Sin(175°) | 0.0872 | | Sin(176°) | 0.0698 | | Sin(177°) | 0.0523 | | Sin(178°) | 0.0349 | | Sin(179°) | 0.0175 | | Sin(180°) | 0 |
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| Sin(181°) | -0.0175 | | Sin(182°) | -0.0349 | | Sin(183°) | -0.0523 | | Sin(184°) | -0.0698 | | Sin(185°) | -0.0872 | | Sin(186°) | -0.1045 | | Sin(187°) | -0.1219 | | Sin(188°) | -0.1392 | | Sin(189°) | -0.1564 | | Sin(190°) | -0.1736 | | Sin(191°) | -0.1908 | | Sin(192°) | -0.2079 | | Sin(193°) | -0.225 | | Sin(194°) | -0.2419 | | Sin(195°) | -0.2588 | | Sin(196°) | -0.2756 | | Sin(197°) | -0.2924 | | Sin(198°) | -0.309 | | Sin(199°) | -0.3256 | | Sin(200°) | -0.342 | | Sin(201°) | -0.3584 | | Sin(202°) | -0.3746 | | Sin(203°) | -0.3907 | | Sin(204°) | -0.4067 | | Sin(205°) | -0.4226 | | Sin(206°) | -0.4384 | | Sin(207°) | -0.454 | | Sin(208°) | -0.4695 | | Sin(209°) | -0.4848 | | Sin(210°) | -0.5 | | Sin(211°) | -0.515 | | Sin(212°) | -0.5299 | | Sin(213°) | -0.5446 | | Sin(214°) | -0.5592 | | Sin(215°) | -0.5736 | | Sin(216°) | -0.5878 |
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| Sin(217°) | -0.6018 | | Sin(218°) | -0.6157 | | Sin(219°) | -0.6293 | | Sin(220°) | -0.6428 | | Sin(221°) | -0.6561 | | Sin(222°) | -0.6691 | | Sin(223°) | -0.682 | | Sin(224°) | -0.6947 | | Sin(225°) | -0.7071 | | Sin(226°) | -0.7193 | | Sin(227°) | -0.7314 | | Sin(228°) | -0.7431 | | Sin(229°) | -0.7547 | | Sin(230°) | -0.766 | | Sin(231°) | -0.7771 | | Sin(232°) | -0.788 | | Sin(233°) | -0.7986 | | Sin(234°) | -0.809 | | Sin(235°) | -0.8192 | | Sin(236°) | -0.829 | | Sin(237°) | -0.8387 | | Sin(238°) | -0.848 | | Sin(239°) | -0.8572 | | Sin(240°) | -0.866 | | Sin(241°) | -0.8746 | | Sin(242°) | -0.8829 | | Sin(243°) | -0.891 | | Sin(244°) | -0.8988 | | Sin(245°) | -0.9063 | | Sin(246°) | -0.9135 | | Sin(247°) | -0.9205 | | Sin(248°) | -0.9272 | | Sin(249°) | -0.9336 | | Sin(250°) | -0.9397 | | Sin(251°) | -0.9455 | | Sin(252°) | -0.9511 |
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| Sin(253°) | -0.9563 | | Sin(254°) | -0.9613 | | Sin(255°) | -0.9659 | | Sin(256°) | -0.9703 | | Sin(257°) | -0.9744 | | Sin(258°) | -0.9781 | | Sin(259°) | -0.9816 | | Sin(260°) | -0.9848 | | Sin(261°) | -0.9877 | | Sin(262°) | -0.9903 | | Sin(263°) | -0.9925 | | Sin(264°) | -0.9945 | | Sin(265°) | -0.9962 | | Sin(266°) | -0.9976 | | Sin(267°) | -0.9986 | | Sin(268°) | -0.9994 | | Sin(269°) | -0.9998 | | Sin(270°) | -1 | | Sin(271°) | -0.9998 | | Sin(272°) | -0.9994 | | Sin(273°) | -0.9986 | | Sin(274°) | -0.9976 | | Sin(275°) | -0.9962 | | Sin(276°) | -0.9945 | | Sin(277°) | -0.9925 | | Sin(278°) | -0.9903 | | Sin(279°) | -0.9877 | | Sin(280°) | -0.9848 | | Sin(281°) | -0.9816 | | Sin(282°) | -0.9781 | | Sin(283°) | -0.9744 | | Sin(284°) | -0.9703 | | Sin(285°) | -0.9659 | | Sin(286°) | -0.9613 | | Sin(287°) | -0.9563 | | Sin(288°) | -0.9511 |
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| Sin(289°) | -0.9455 | | Sin(290°) | -0.9397 | | Sin(291°) | -0.9336 | | Sin(292°) | -0.9272 | | Sin(293°) | -0.9205 | | Sin(294°) | -0.9135 | | Sin(295°) | -0.9063 | | Sin(296°) | -0.8988 | | Sin(297°) | -0.891 | | Sin(298°) | -0.8829 | | Sin(299°) | -0.8746 | | Sin(300°) | -0.866 | | Sin(301°) | -0.8572 | | Sin(302°) | -0.848 | | Sin(303°) | -0.8387 | | Sin(304°) | -0.829 | | Sin(305°) | -0.8192 | | Sin(306°) | -0.809 | | Sin(307°) | -0.7986 | | Sin(308°) | -0.788 | | Sin(309°) | -0.7771 | | Sin(310°) | -0.766 | | Sin(311°) | -0.7547 | | Sin(312°) | -0.7431 | | Sin(313°) | -0.7314 | | Sin(314°) | -0.7193 | | Sin(315°) | -0.7071 | | Sin(316°) | -0.6947 | | Sin(317°) | -0.682 | | Sin(318°) | -0.6691 | | Sin(319°) | -0.6561 | | Sin(320°) | -0.6428 | | Sin(321°) | -0.6293 | | Sin(322°) | -0.6157 | | Sin(323°) | -0.6018 | | Sin(324°) | -0.5878 |
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| Sin(325°) | -0.5736 | | Sin(326°) | -0.5592 | | Sin(327°) | -0.5446 | | Sin(328°) | -0.5299 | | Sin(329°) | -0.515 | | Sin(330°) | -0.5 | | Sin(331°) | -0.4848 | | Sin(332°) | -0.4695 | | Sin(333°) | -0.454 | | Sin(334°) | -0.4384 | | Sin(335°) | -0.4226 | | Sin(336°) | -0.4067 | | Sin(337°) | -0.3907 | | Sin(338°) | -0.3746 | | Sin(339°) | -0.3584 | | Sin(340°) | -0.342 | | Sin(341°) | -0.3256 | | Sin(342°) | -0.309 | | Sin(343°) | -0.2924 | | Sin(344°) | -0.2756 | | Sin(345°) | -0.2588 | | Sin(346°) | -0.2419 | | Sin(347°) | -0.225 | | Sin(348°) | -0.2079 | | Sin(349°) | -0.1908 | | Sin(350°) | -0.1736 | | Sin(351°) | -0.1564 | | Sin(352°) | -0.1392 | | Sin(353°) | -0.1219 | | Sin(354°) | -0.1045 | | Sin(355°) | -0.0872 | | Sin(356°) | -0.0698 | | Sin(357°) | -0.0523 | | Sin(358°) | -0.0349 | | Sin(359°) | -0.0175 | | Sin(360°) | -0 |
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