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08 March, 2023

त्रिकोणमिति फॉर्मूला

Mr. B. R. Gupta (Govt. Teacher)
M.A., B.ED., CTET, STET

 

Trigonometric functions

sin α,    cos α
tan α = sin α,   α ≠ π + πn,   n є Z
cos α2
cot α = cos α,   α ≠ π + πn,   n є Z
sin α
tan α · cot α = 1
sec α = 1,   α ≠ π + πn,   n є Z
cos α2
cosec α = 1,   α ≠ π + πn,   n є Z
sin α

Pythagorean identity

sin2 α + cos2 α = 1
1 + tan2 α = 1
cos2 α
1 + cot2 α = 1
sin2 α

Sum-Difference Formulas

sin(α + β) = sin α · cos β + cos α · sin β
sin(α – β) = sin α · cos β – cos α · sin β
cos(α + β) = cos α · cos β – sin α · sin β
cos(α – β) = cos α · cos β + sin α · sin β
tan(α + β) = tan α + tan β
1 – tanα · tan β
tan(α – β) = tan α – tan β
1 + tanα · tan β
cot(α + β) = cotα · cot β - 1
cot β + cot α
cot(α - β) = cotα · cot β + 1
cot β - cot α

Double angle formulas

sin 2α = 2 sin α · cos α
cos 2α = cos2 α - sin2 α
tan 2α = 2 tan α
1 - tan2 α
cot 2α = cot2 α - 1
2 cot α

Triple angle formulas

sin 3α = 3 sin α - 4 sin3 α
cos 3α = 4 cos3 α - 3 cos α
tan 3α = 3 tan α - tan3 α
1 - 3 tan2 α
cot 3α = 3 cot α - cot3 α
1 - 3 cot2 α

Power-reduction formula

sin2 α = 1 - cos 2α
2
cos2 α = 1 + cos 2α
2
sin3 α = 3 sin α - sin 3α
4
cos3 α = 3 cos α + cos 3α
4

Sum (difference) to product formulas

sin α + sin β = 2 sin α + β cos α - β
22
sin α - sin β = 2 sin α - β cos α + β
22
cos α + cos β = 2 cos α + β cos α - β
22
cos α - cos β = -2 sin α + β sin α - β
22
tan α + sin β =  sin(α + β)
cos α · cos β
tan α - sin β =  sin(α - β)
cos α · cos β
cot α + sin β =  sin(α + β)
sin α · sin β
cot α - sin β =  sin(α - β)
sin α · sin β
a sin α + b cos α = r sin (α + φ),

where r2 = a2 + b2, sin φ = b , tan φ = b
ra

Product to sum (difference) formulas

sin α · sin β = 1(cos(α - β) - cos(α + β))
2
sin α · cos β = 1(sin(α + β) + sin(α - β))
2
cos α · cos β = 1(cos(α + β) + cos(α - β))
2

Tangent half-angle substitution

sin α = 2 tan (α/2)
1 + tan2 (α/2)
cos α = 1 - tan2 (α/2)
1 + tan2 (α/2)
tan α = 2 tan (α/2)
1 - tan2 (α/2)
cot α = 1 - tan2 (α/2)
2 tan (α/2)

Table of cosine in radians

α0π6π4π3π2π3π22π
cos α13222120-101

Table of angles cosines from 0° to 180°

cos(0°) = 1
cos(1°) = 0.999848
cos(2°) = 0.999391
cos(3°) = 0.99863
cos(4°) = 0.997564
cos(5°) = 0.996195
cos(6°) = 0.994522
cos(7°) = 0.992546
cos(8°) = 0.990268
cos(9°) = 0.987688
cos(10°) = 0.984808
cos(11°) = 0.981627
cos(12°) = 0.978148
cos(13°) = 0.97437
cos(14°) = 0.970296
cos(15°) = 0.965926
cos(16°) = 0.961262
cos(17°) = 0.956305
cos(18°) = 0.951057
cos(19°) = 0.945519
cos(20°) = 0.939693
cos(21°) = 0.93358
cos(22°) = 0.927184
cos(23°) = 0.920505
cos(24°) = 0.913545
cos(25°) = 0.906308
cos(26°) = 0.898794
cos(27°) = 0.891007
cos(28°) = 0.882948
cos(29°) = 0.87462
cos(30°) = 0.866025
cos(31°) = 0.857167
cos(32°) = 0.848048
cos(33°) = 0.838671
cos(34°) = 0.829038
cos(35°) = 0.819152
cos(36°) = 0.809017
cos(37°) = 0.798636
cos(38°) = 0.788011
cos(39°) = 0.777146
cos(40°) = 0.766044
cos(41°) = 0.75471
cos(42°) = 0.743145
cos(43°) = 0.731354
cos(44°) = 0.71934
cos(45°) = 0.707107
cos(46°) = 0.694658
cos(47°) = 0.681998
cos(48°) = 0.669131
cos(49°) = 0.656059
cos(50°) = 0.642788
cos(51°) = 0.62932
cos(52°) = 0.615661
cos(53°) = 0.601815
cos(54°) = 0.587785
cos(55°) = 0.573576
cos(56°) = 0.559193
cos(57°) = 0.544639
cos(58°) = 0.529919
cos(59°) = 0.515038
cos(60°) = 0.5
cos(61°) = 0.48481
cos(62°) = 0.469472
cos(63°) = 0.45399
cos(64°) = 0.438371
cos(65°) = 0.422618
cos(66°) = 0.406737
cos(67°) = 0.390731
cos(68°) = 0.374607
cos(69°) = 0.358368
cos(70°) = 0.34202
cos(71°) = 0.325568
cos(72°) = 0.309017
cos(73°) = 0.292372
cos(74°) = 0.275637
cos(75°) = 0.258819
cos(76°) = 0.241922
cos(77°) = 0.224951
cos(78°) = 0.207912
cos(79°) = 0.190809
cos(80°) = 0.173648
cos(81°) = 0.156434
cos(82°) = 0.139173
cos(83°) = 0.121869
cos(84°) = 0.104528
cos(85°) = 0.087156
cos(86°) = 0.069756
cos(87°) = 0.052336
cos(88°) = 0.034899
cos(89°) = 0.017452
cos(90°) = 0
cos(91°) = -0.017452
cos(92°) = -0.034899
cos(93°) = -0.052336
cos(94°) = -0.069756
cos(95°) = -0.087156
cos(96°) = -0.104528
cos(97°) = -0.121869
cos(98°) = -0.139173
cos(99°) = -0.156434
cos(100°) = -0.173648
cos(101°) = -0.190809
cos(102°) = -0.207912
cos(103°) = -0.224951
cos(104°) = -0.241922
cos(105°) = -0.258819
cos(106°) = -0.275637
cos(107°) = -0.292372
cos(108°) = -0.309017
cos(109°) = -0.325568
cos(110°) = -0.34202
cos(111°) = -0.358368
cos(112°) = -0.374607
cos(113°) = -0.390731
cos(114°) = -0.406737
cos(115°) = -0.422618
cos(116°) = -0.438371
cos(117°) = -0.45399
cos(118°) = -0.469472
cos(119°) = -0.48481
cos(120°) = -0.5
cos(121°) = -0.515038
cos(122°) = -0.529919
cos(123°) = -0.544639
cos(124°) = -0.559193
cos(125°) = -0.573576
cos(126°) = -0.587785
cos(127°) = -0.601815
cos(128°) = -0.615661
cos(129°) = -0.62932
cos(130°) = -0.642788
cos(131°) = -0.656059
cos(132°) = -0.669131
cos(133°) = -0.681998
cos(134°) = -0.694658
cos(135°) = -0.707107
cos(136°) = -0.71934
cos(137°) = -0.731354
cos(138°) = -0.743145
cos(139°) = -0.75471
cos(140°) = -0.766044
cos(141°) = -0.777146
cos(142°) = -0.788011
cos(143°) = -0.798636
cos(144°) = -0.809017
cos(145°) = -0.819152
cos(146°) = -0.829038
cos(147°) = -0.838671
cos(148°) = -0.848048
cos(149°) = -0.857167
cos(150°) = -0.866025
cos(151°) = -0.87462
cos(152°) = -0.882948
cos(153°) = -0.891007
cos(154°) = -0.898794
cos(155°) = -0.906308
cos(156°) = -0.913545
cos(157°) = -0.920505
cos(158°) = -0.927184
cos(159°) = -0.93358
cos(160°) = -0.939693
cos(161°) = -0.945519
cos(162°) = -0.951057
cos(163°) = -0.956305
cos(164°) = -0.961262
cos(165°) = -0.965926
cos(166°) = -0.970296
cos(167°) = -0.97437
cos(168°) = -0.978148
cos(169°) = -0.981627
cos(170°) = -0.984808
cos(171°) = -0.987688
cos(172°) = -0.990268
cos(173°) = -0.992546
cos(174°) = -0.994522
cos(175°) = -0.996195
cos(176°) = -0.997564
cos(177°) = -0.99863
cos(178°) = -0.999391
cos(179°) = -0.999848
cos(180°) = -1

Table of angles cosines from 181° to 360°

cos(181°) = -0.999848
cos(182°) = -0.999391
cos(183°) = -0.99863
cos(184°) = -0.997564
cos(185°) = -0.996195
cos(186°) = -0.994522
cos(187°) = -0.992546
cos(188°) = -0.990268
cos(189°) = -0.987688
cos(190°) = -0.984808
cos(191°) = -0.981627
cos(192°) = -0.978148
cos(193°) = -0.97437
cos(194°) = -0.970296
cos(195°) = -0.965926
cos(196°) = -0.961262
cos(197°) = -0.956305
cos(198°) = -0.951057
cos(199°) = -0.945519
cos(200°) = -0.939693
cos(201°) = -0.93358
cos(202°) = -0.927184
cos(203°) = -0.920505
cos(204°) = -0.913545
cos(205°) = -0.906308
cos(206°) = -0.898794
cos(207°) = -0.891007
cos(208°) = -0.882948
cos(209°) = -0.87462
cos(210°) = -0.866025
cos(211°) = -0.857167
cos(212°) = -0.848048
cos(213°) = -0.838671
cos(214°) = -0.829038
cos(215°) = -0.819152
cos(216°) = -0.809017
cos(217°) = -0.798636
cos(218°) = -0.788011
cos(219°) = -0.777146
cos(220°) = -0.766044
cos(221°) = -0.75471
cos(222°) = -0.743145
cos(223°) = -0.731354
cos(224°) = -0.71934
cos(225°) = -0.707107
cos(226°) = -0.694658
cos(227°) = -0.681998
cos(228°) = -0.669131
cos(229°) = -0.656059
cos(230°) = -0.642788
cos(231°) = -0.62932
cos(232°) = -0.615661
cos(233°) = -0.601815
cos(234°) = -0.587785
cos(235°) = -0.573576
cos(236°) = -0.559193
cos(237°) = -0.544639
cos(238°) = -0.529919
cos(239°) = -0.515038
cos(240°) = -0.5
cos(241°) = -0.48481
cos(242°) = -0.469472
cos(243°) = -0.45399
cos(244°) = -0.438371
cos(245°) = -0.422618
cos(246°) = -0.406737
cos(247°) = -0.390731
cos(248°) = -0.374607
cos(249°) = -0.358368
cos(250°) = -0.34202
cos(251°) = -0.325568
cos(252°) = -0.309017
cos(253°) = -0.292372
cos(254°) = -0.275637
cos(255°) = -0.258819
cos(256°) = -0.241922
cos(257°) = -0.224951
cos(258°) = -0.207912
cos(259°) = -0.190809
cos(260°) = -0.173648
cos(261°) = -0.156434
cos(262°) = -0.139173
cos(263°) = -0.121869
cos(264°) = -0.104528
cos(265°) = -0.087156
cos(266°) = -0.069756
cos(267°) = -0.052336
cos(268°) = -0.034899
cos(269°) = -0.017452
cos(270°) = -0
cos(271°) = 0.017452
cos(272°) = 0.034899
cos(273°) = 0.052336
cos(274°) = 0.069756
cos(275°) = 0.087156
cos(276°) = 0.104528
cos(277°) = 0.121869
cos(278°) = 0.139173
cos(279°) = 0.156434
cos(280°) = 0.173648
cos(281°) = 0.190809
cos(282°) = 0.207912
cos(283°) = 0.224951
cos(284°) = 0.241922
cos(285°) = 0.258819
cos(286°) = 0.275637
cos(287°) = 0.292372
cos(288°) = 0.309017
cos(289°) = 0.325568
cos(290°) = 0.34202
cos(291°) = 0.358368
cos(292°) = 0.374607
cos(293°) = 0.390731
cos(294°) = 0.406737
cos(295°) = 0.422618
cos(296°) = 0.438371
cos(297°) = 0.45399
cos(298°) = 0.469472
cos(299°) = 0.48481
cos(300°) = 0.5
cos(301°) = 0.515038
cos(302°) = 0.529919
cos(303°) = 0.544639
cos(304°) = 0.559193
cos(305°) = 0.573576
cos(306°) = 0.587785
cos(307°) = 0.601815
cos(308°) = 0.615661
cos(309°) = 0.62932
cos(310°) = 0.642788
cos(311°) = 0.656059
cos(312°) = 0.669131
cos(313°) = 0.681998
cos(314°) = 0.694658
cos(315°) = 0.707107
cos(316°) = 0.71934
cos(317°) = 0.731354
cos(318°) = 0.743145
cos(319°) = 0.75471
cos(320°) = 0.766044
cos(321°) = 0.777146
cos(322°) = 0.788011
cos(323°) = 0.798636
cos(324°) = 0.809017
cos(325°) = 0.819152
cos(326°) = 0.829038
cos(327°) = 0.838671
cos(328°) = 0.848048
cos(329°) = 0.857167
cos(330°) = 0.866025
cos(331°) = 0.87462
cos(332°) = 0.882948
cos(333°) = 0.891007
cos(334°) = 0.898794
cos(335°) = 0.906308
cos(336°) = 0.913545
cos(337°) = 0.920505
cos(338°) = 0.927184
cos(339°) = 0.93358
cos(340°) = 0.939693
cos(341°) = 0.945519
cos(342°) = 0.951057
cos(343°) = 0.956305
cos(344°) = 0.961262
cos(345°) = 0.965926
cos(346°) = 0.970296
cos(347°) = 0.97437
cos(348°) = 0.978148
cos(349°) = 0.981627
cos(350°) = 0.984808
cos(351°) = 0.987688
cos(352°) = 0.990268
cos(353°) = 0.992546
cos(354°) = 0.994522
cos(355°) = 0.996195
cos(356°) = 0.997564
cos(357°) = 0.99863
cos(358°) = 0.999391
cos(359°) = 0.999848
cos(360°) = 1

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